Double Bruhat Cells and Total Positivity

JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY Volume 12, Number 2, April 1999, Pages 335–380 S 0894-0347(99)00295-7

DOUBLE BRUHAT CELLS AND TOTAL POSITIVITY

SERGEY FOMIN AND ANDREI ZELEVINSKY

Contents 0. Introduction 336 1. Main results 337 1.1. Semisimple groups 337 1.2. Factorization problem 338 1.3. Total positivity 339 1.4. Generalized minors 339 1.5. The twist maps 340 1.6. Formulas for factorization parameters 341 1.7. Total positivity criteria 343 1.8. Fundamental determinantal identities 343 2. Preliminaries 344 2.1. Involutions 344 2.2. Commutation relations 345 2.3. Generalized determinantal identities 346 2.4. Aﬃne coordinates in Schubert cells 349 2.5. y-coordinates in double Bruhat cells 351 2.6. Factorization problem in Schubert cells 353 2.7. Totally positive bases for N (w) 354 2.8. Total positivity in y-coordinates− 356 3. Proofs of the main results 357 3.1. Proofs of Theorems 1.1, 1.2, and 1.3 357 3.2. Proofs of Theorems 1.6 and 1.7 359 3.3. Proof of Theorem 1.9 360 3.4. Proofs of Theorems 1.11 and 1.12 363 4. GLn theory 364 4.1. Bruhat cells and double Bruhat cells for GLn 364 4.2. Factorization problem for GLn 365 4.3. The twist maps for GLn 367 4.4. Double pseudoline arrangements 369 4.5. Solution to the factorization problem 371 4.6. Applications to total positivity 374 References 379

Received by the editors February 12, 1998. 1991 Mathematics Subject Classiﬁcation. Primary 22E46; Secondary 05E15, 15A23. Key words and phrases. Total positivity, semisimple groups, Bruhat decomposition. The authors were supported in part by NSF grants #DMS-9400914, #DMS-9625511, and #DMS-9700927, and by MSRI (NSF grant #DMS-9022140).

c 1999 by Sergey Fomin and Andrei Zelevinsky

335

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0. Introduction This paper continues the algebraic study of total positivity in semisimple algebraic groups undertaken in [3, 4]. Traditional theory of total positivity, pioneered in the 1930s by Gantmacher, Krein, and Schoenberg, studies matrices whose all minors are nonnegative. Recently, G. Lusztig [18] extended this classical subject by introducing the totally nonnegative variety G 0 in an arbitrary reductive group G. (Lusztig’s study was motivated by surprising≥ connections he discovered between total positivity and his theory of canonical bases for quantum groups.) The main object of study in [3, 4] was the structure of the intersection G 0 N,whereNis a maximal unipotent subgroup in G. In this paper we extend≥ the∩ results of [3, 4] to the whole variety G 0. ≥ It turns out that the natural geometric framework for the study of G 0 is provided by the decomposition of G into the disjoint union of double Bruhat≥ cells Gu,v = BuB B vB ;hereBand B are two opposite Borel subgroups in G, and u and v belong∩ − to− the Weyl group−W of G. We believe these double cells to be a very interesting object of study in their own right. The term “cells” might be misleading: in fact, the topology of Gu,v is in general quite nontrivial. (In some special cases, the “real part” of Gu,v was studied in [21, 22]. V. Deodhar [9] studied the intersections BuB B vB whose properties are very different from those of Gu,v.) ∩ − We study a family of birational parametrizations of Gu,v, one for each reduced expression i of the element (u, v) in the Coxeter group W W .Everysuch parametrization can be thought of as a system of local coordinates× in Gu,v.We call these coordinates the factorization parameters associated to i. They are obtained by expressing a generic element x Gu,v as an element of the maximal torus H = B B multiplied by the product∈ of elements of various one-parameter subgroups in ∩G associated− with simple roots and their negatives; the reduced expression i prescribes the order of factors in this product. The main technical result of this paper (Theorem 1.9) is an explicit formula for these factorization parameters as rational functions on the double Bruhat cell Gu,v. Theorem 1.9 is formulated in terms of a special family of regular functions ∆γ,δ on the group G. These functions are suitably normalized matrix coefficients corresponding to pairs of extremal weights (γ,δ) in some fundamental representation of G. We believe these functions are very interesting subjects that merit further study. For the type A, they specialize to the minors of a matrix, and their properties are of course developed in great detail. It would be very interesting to extend the main body of the classical theory of determinantal identities to the family of functions ∆γ,δ. In this paper, we make the first steps in this direction (see especially Theorems 1.16 and 1.17 below). As in [3, 4], the main algebraic relations involving factorization parameters and generalized minors ∆γ,δ can be written in a “subtraction-free” form, and thus the theory can be developed over an arbitrary semifield. The readers familiar with [3, 4] will have no trouble extending the corresponding results there (cf. [3, Section 2]) to the more general context of this paper. We do not pursue this path here since at the moment we have not developed applications of this more general setup. Returning to total positivity, our explicit formulas for factorization parameters allow us to obtain a family of total positivity criteria, each of which efficiently tests whether a given element x from an arbitrary double Bruhat cell Gu,v is totally

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nonnegative. More specifically, each of our criteria consists in verifying whether x satisfies a system of inequalities of the form ∆γ,δ(x) > 0, the number of these inequalities being equal to the dimension of Gu,v (see Theorem 1.11). In Section 1 we give precise formulations of our main results. Their proofs are given in Sections 2 and 3. The last Section 4 contains applications of our theory to the case of the general linear group.ThecaseG=GLn is treated separately for a number of reasons. First, GLn is not a semisimple group (although everything reduces easily to SLn). Fur- thermore, the questions that we consider become some very natural linear-algebraic questions whose understanding does not require any Lie-theoretic background. For instance, the factorization parameters become the parameters in factorizations of a square matrix into the smallest possible number of elementary Jacobi matrices. Our main results seem to be new even in this case. In Section 4, we tried to present them in an elementary form, making this section as self-contained as possible. Last but not least, our results in the GLn case have a particularly transparent formulation in the language of pseudoline arrangements. The criteria of Theorem 1.11 lead us to new solutions of the classical problem of efficiently testing whether a given n n matrix is totally positive, i.e., has all minors > 0. This problem has a long× history. The ground was broken in 1912 by M. Fekete [10] who proved that positivity of all solid minors, i.e., those formed by several consecutive rows and several consecutive columns, is sufficient for total positivity. It took a while before it was realized that Fekete’s criterion was far from optimal: as shown in [13], it is enough to check the positivity of those solid minors that involve the first row or the first column of the matrix (this result can also be derived from [8]). Note that the number of such minors is n2 while the total 2n 2 number of minors is n 1. One can show that at least n minors are needed to characterize total positivity;− thus the criterion reproduced above is “minimal”. Theorem 4.13 (a specialization of Theorem 1.11) includes this criterion into a family of minimal total positivity criteria associated to shuffles of two reduced words for the permutation wo = nn 1 2 1. For example, for n = 3 we obtain 34 different criteria shown in Figure 8− at the··· end of the paper.

1. Main results 1.1. Semisimple groups. We begin by introducing general terminology and notation (mostly standard) for semisimple Lie groups and algebras (cf., e.g., [23]). Let g be a semisimple complex Lie algebra of rank r with the Cartan decomposition g = n h n.Letei,hi,fi ,fori=1,... ,r, be the standard generators of g, − ⊕ ⊕ and let A =(aij )betheCartan matrix.Thusaij = αj (hi), where α1,... ,αr h∗ are the simple roots of g.LetGbe a simply connected complex Lie group with∈ the Lie algebra g.LetN,Hand N be closed subgroups of G with Lie algebras n , h and n, respectively.− Thus H is a maximal torus,andNand N are two opposite− maximal unipotent subgroups of G.LetB =HN and B =− HN be − − the corresponding pair of opposite Borel subgroups.Fori=1,...,r and t C,we write ∈

(1.1) xi(t)=exp(tei) ,xi(t)=exp(tfi) ,

so that t xi(t)(resp.t xi(t)) is a one-parameter subgroup in N (resp. in N ). 7→ 7→ − We prefer the notation xi(t) to the usual yi(t), for reasons that will become clear later. It will be convenient to denote [1,r]= 1,... ,r and [1, r]= 1,... ,r . { } { }

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The weight lattice P is the set of all weights γ h∗ such that γ(hi) Z for all i. ∈ ∈ The group P has a Z-basis formed by the fundamental weights ω1,... ,ωr defined by ω (h )=δ .Everyweightγ Pgives rise to a multiplicative character a aγ i j ij ∈ 7→ of the maximal torus H; this character is given by exp(h)γ = eγ(h) (h h). ∈ The Weyl group W of G is defined by W = NormG(H)/H. The action of W on H by conjugation gives rise to the action of W on the weight lattice P given by w(γ) 1 γ (1.2) a =(w− aw) (w W, a H, γ P) . ∈ ∈ ∈ As usual, we identify W with the corresponding group of linear transformations of h∗. The group W is a Coxeter group generated by simple reflections s1,... ,sr given by s (γ)=γ γ(h)α ,forγ h∗. i − i i ∈ Areduced word for w W is a sequence of indices i =(i1,... ,i ) of shortest ∈ m possible length m such that w = si1 sim .Thenumbermis denoted by `(w) and is called the length of w. The set··· of reduced words for w will be denoted by R(w). The Weyl group W has the unique element wo of maximal length, and 1 `(wo)=`(w)+`(w− wo) for any w W . ∈ 1.2. Factorization problem. Recall that the group G has two Bruhat decompositions, with respect to opposite Borel subgroups B and B : − G = BuB = B vB . − − u W v W [∈ [∈ The double Bruhat cells Gu,v are defined by Gu,v = BuB B vB ; ∩ − − thus G is the disjoint union of all Gu,v for (u, v) W W . ∈ × Theorem 1.1. The variety Gu,v is biregularly isomorphic to a Zariski open subset of an affine space of dimension r + `(u)+`(v). We will study a family of birational parametrizations of Gu,v. To describe these parametrizations, we will need the following combinatorial notion. A double reduced word for the elements u, v W is a reduced word for an element (u, v) of the Coxeter group W W . To avoid∈ confusion, we will use the indices 1, 2,... ,r for the simple reflections× in the first copy of W ,and1,2,... ,r for the second copy. A double reduced word for (u, v) is nothing but a shuffle of a reduced word for u written in the alphabet [1, r] and a reduced word for v writteninthe alphabet [1,r]. We denote the set of double reduced words for (u, v)byR(u, v). For any sequence i =(i1,... ,im) of indices from the alphabet [1,r] [1,r], m ∪ consider the map xi : H C G defined by × → (1.3) x (a; t1,... ,t )=ax (t1) x (t ) , i m i1 ··· im m where we use the notation of (1.1). Let C=0 denote the set of nonzero complex numbers. 6

Theorem 1.2. For any u, v W and i =(i1,... ,im) R(u, v),themapxi ∈ m ∈ restricts to a biregular isomorphism between H C=0 and a Zariski open subset of the double Bruhat cell Gu,v. × 6

m u,v Thus xi gives rise to a birational isomorphism between H C and G .We remark that this property holds if and only if i is a double reduced× word for (u, v).

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u,v Theorem 1.2 tells us that for a generic element x G and any i =(i1,... ,i ) ∈ m R(u, v), there are uniquely deﬁned factorization parameters a, t1,...,t such that ∈ m

(1.4) x = ax (t1) x (t ) . i1 ··· im m One of our main results is the solution of the following factorization problem: ﬁnd 1 u,v explicit formulas for the inverse birational isomorphism xi− between G and H Cm. In other words, we express the factorization parameters in terms of the× element x and the double reduced word i underlying the factorization. Our solution of the factorization problem generalizes Theorems 1.4 and 6.2 in [4] (the case of x N), which in turn generalize Theorems 1.4 and 5.4.2 in [3] (same, for type A). ∈

1.3. Total positivity. We will apply our solution of the factorization problem to the study of total positivity. Following G. Lusztig [18], let us define totally nonnegative elements in G.LetH>0be the subgroup of H consisting of all a γ ∈ H such that a R>0 for any weight γ P . (We denote by R>0 the set of ∈ ∈ positive reals.) The set G 0 of totally nonnegative elements is, by definition, the ≥ multiplicative semigroup in G generated by H>0 and the elements xi(t)andxi(t), for i [1,r]andt R>0. It is easy to see that a totally nonnegative element ∈ ∈ x G can be represented as x = x (a; t1,... ,t ), for some sequence i, with all ∈ i m the tk positive and a H>0 .ForthetypeAr(i.e., for G = SLn(C), n = r +1), a theorem in [17], based∈ on a result by A. Whitney [24], tells us that the above definition of total nonnegativity coincides with the usual one [2, 16]: a matrix (with determinant 1) is totally nonnegative if and only if all its minors are nonnegative. u,v The set G 0 is the disjoint union of the subsets G>0 obtained by intersecting it with double Bruhat≥ cells:

u,v u,v G>0 = G 0 G . ≥ ∩ u,v We call the G>0 totally positive varieties; they will be one of the main objects of study in this paper. The terminology is justiﬁed by the following observation made by Lusztig [18]: in the special case G = SLn(C)andu=v=wo,the u,v variety G>0 is the set of all n n-matrices (with determinant 1) which are (strictly) totally positive in the usual sense,× i.e., all their minors are positive. We note that u,v the decomposition G 0 = G>0 appeared in [18] (without explicit mentioning of double Bruhat cells).≥ The following theorem can be derived from the results in [18]. S Theorem 1.3. For any u, v W and any double reduced word i R(u, v),the ∈ m u,v ∈ map xi restricts to a bijection H>0 R 0 G 0 . × > → > Informally speaking, Theorem 1.3 asserts that an element x Gu,v is totally nonnegative if and only if for some (equivalently, any) double reduced∈ word i ∈ R(u, v), the factorization parameters a, t1,...,tm appearing in (1.4) are well deﬁned and positive. Thus the solution of the factorization problem will lead to a family of total positivity criteria—one for each double reduced word.

1.4. Generalized minors. The main ingredients of our answer to the factorization problem are similar to those in [4]: a family of regular functions on G generalizing 1 1 minors of a square matrix, and a biregular “twist” Gu,v Gu− ,v− . →

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We will denote by G0 = N HN the set of elements x G that have Gaussian decomposition; this decomposition− will be written as ∈

(1.5) x =[x] [x]0[x]+ . − ωi Following [4], for any fundamental weight ωi ,let∆ denote the regular function on G whose restriction to the open set G0 is given by

ωi ωi (1.6) ∆ (x)=[x]0 .

ωi For the type Ar ,the∆ (x) are the principal minors of a matrix x. We will use the same terminology in the general case as well. To define the analogues of arbitrary minors, we will need two special representatives w, w G for any element w W . For a simple reflection s ,set ∈ ∈ i 0 1 01 s =ϕ ,s=ϕ , i i 10− i i 10 − where ϕ : SL2 G is the group homomorphism given by i → 1 t 10 (1.7) ϕ =x(t),ϕ =x(t). i 01 i i t1 i Alternatively, we could define (1.8) s = x ( 1)x (1)x ( 1) , s = x (1)x ( 1)x (1) . i i − i i − i i i − i It is known (and easy to check) that the families si and si satisfy the braid relations in W . It follows that the representatives{w }and w{can} be uniquely and unambiguously defined for any w W by the condition that ∈ (1.9) w w = w w , w w = w w 0 00 0 · 00 0 00 0 · 00 whenever `(w0w00)=`(w0)+`(w00). Definition 1.4. For u, v W , define a regular function ∆ on G by setting ∈ uωi,vωi ωi 1 (1.10) ∆uωi,vωi (x)=∆ u− xv . One has to check that this is well defined, i.e., the right-hand side of (1.10) only depends on the weights uωi and vωi, not on the particular choice of u and v.This is done in Section 2.3 (cf. Proposition 2.3).

For the type Ar , the functions ∆uωi,vωi (x) are the minors of a matrix x.Inthe general case, we will refer to them as generalized minors, or simply as minors if there will be no danger of confusion. 1.5. The twist maps. To deﬁne the twist maps, we will need the involutive automorphism x xθ of the group G which is uniquely determined by 7→ θ 1 θ θ (1.11) a = a− (a H) ,x(t)=x(t),x(t)=x(t). ∈ i i i i Notice that the involution θ preserves total nonnegativity. For the type Ar ,ifx is a matrix with determinant 1, then the matrix xθ is formed by signless cofactors of x; in other words, the (i, j)-entry of xθ is simply the minor of x obtained by deleting the ith row and the jth column.

u,v Deﬁnition 1.5. For any u, v W ,thetwistmapζ : x x0 is deﬁned by ∈ 7→ θ 1 1 1 1 1 1 (1.12) x0 = [u− x]− u− x v− [xv− ]+− . −

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Theorem 1.6. The right-hand side of (1.12) is well defined for any x Gu,v,and ∈ 1 1 the twist map ζu,v establishes a biregular isomorphism between Gu,v and Gu− ,v− . 1 1 The inverse isomorphism is ζu− ,v− . The specific choice of representatives for u and v in (1.12) is essential for the following important property. 1 1 Theorem 1.7. The twist map ζu,v restricts to a bijection Gu,v Gu− ,v− . >0 → >0 Example 1.8. Let G = SL2(C). Then W = S2 consists of two permutations: e and wo ;thusGis partitioned into four double Bruhat cells. Table 1 shows the x x conditions under which a matrix x = 11 12 with determinant 1 belongs to x21 x22 u,v each of these cells, or to the corresponding totally positive variety G>0 . The table also shows the formulas defining each twist map ζu,v.

Table 1. Double Bruhat decomposition and the twist maps for SL2

u = eu=eu=wo u=wo v=ev=wo v=ev=wo

x12 =0 x12 =0 x12 =0 x12 =0 Gu,v 6 6 x21 =0 x21 =0 x21 =0 x21 =0 6 6 x11 > 0 x11 > 0 x11 > 0 x11 > 0 u,v G>0 x12 =0 x12 > 0 x12 =0 x12 > 0 x21 =0 x21 =0 x21 > 0 x21 > 0 1 1 1 1 1 1 1 u,v x11− 0 x12− x11− x21− 0 x11x12− x21− x21− ζ (x) 1 1 0 x 0 x x− x x− x 11 12 11 21 12 22

1.6. Formulas for factorization parameters. To give explicit formulas for factorization parameters, we will need some more notation. First, we will write (1.13) i = i = i, ε(i)=1,ε(i)= 1, | | | | − for any i [1,r]. Let us ﬁx a pair (u, v) W W and a double reduced word ∈ ∈ × i =(i1,... ,im) R(u, v). Recall that i is a shuﬄe of a reduced word for u written in the alphabet∈ [1, r] and a reduced word for v written in the alphabet [1,r]. In particular, the length m of i is equal to `(u)+`(v). We will add r additional entries im+1,... ,im+r at the end of i by setting

(1.14) i + = j (j [1,r]) . m j ∈ For k [1,m+r], let us denote ∈ (1.15) u k = s i ,vm. (For example, if i = 213321211, then, say, u 7 = s1s2 and ≥ v<7 = s1s3s2 .) Let us deﬁne a regular function ∆k =∆k,i on G by

(1.16) ∆k =∆k,i =∆u kωi ,v

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For k [1,m+r], we denote ∈

k− =max l:l

unless il = ik for all l

1ifklor ε(i ) = ε(i ) .  k 6 l With all this notation in mind, we now formulate our ﬁrst main result: a solution to the factorization problem of Section 1.2.

Theorem 1.9. Let i =(i1,... ,im)be a double reduced word for (u, v), and suppose u,v an element x G can be factored as x = ax (t1) x (t ),witha Hand ∈ i1 ··· im m ∈ all tk nonzero complex numbers. Then the factorization parameters are determined by the following formulas:

m+r (χ(k,l− ) χ(k,l))a i , i (1.18) tk = ∆l,i(x0) − | l| | k| ; =1 Yl

ωi ε(ik) (1.19) a = ∆k,i(x0) , 1 k m+r ≤ Y≤ ik =i, ε(ik)=ε(i ) | | 6 k− u,v where x0 = ζ (x), and we use the convention ε(i0)=1. Formulas (1.19) can be restated as the following closed expression for the element a:

ε(ik ) 1 (1.20) a = u−kx0v

Since (1.18) and (1.19) express the m+ r independent parameters t1,...,tm and ω1 ωr a ,...,a as Laurent monomials in the m+r minors ∆1,i(x0),...,∆m+r,i(x0), we obtain the following important corollary.

Theorem 1.10. Under the assumptions of Theorem 1.9, the parameters t1,...,tm ω1 ωr and a ,...,a are related to the minors ∆1,i(x0),...,∆m+r,i(x0) by an invertible monomial transformation.

The inverse of this monomial transformation can be computed explicitly: one can show that it is given by

1 1 u− u ω (h ) − ( ) uu ω l k ik il v

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1.7. Total positivity criteria. Theorem 1.9 implies a family of criteria for total positivity that generalize the ones in [3, 4]. Each of these criteria asserts that a u,v u,v point x G belongs to the totally positive variety G>0 if and only if a particular collection∈ of `(u)+`(v)+r minors evaluated at x are all positive. For every double reduced word i =(i1,... ,im) R(u, v), let F (i)denotethe following collection of m + r minors: ∈ (1.22) F (i)= ∆ : k [1,m+r] { k,i∗ ∈ } 1 1 (cf. (1.16)), where i∗ =(i ,...,i1) R(u− ,v− )isiwritten backwards. m ∈ Theorem 1.11. Let i R(u, v). An element x Gu,v is totally nonnegative if and only if ∆(x) > 0 for∈ any minor ∆ F (i). ∈ ∈ As a consequence, for any two double reduced words i, i0 R(u, v), the positivity of all minors from F (i)atagivenx Gu,v is equivalent∈ to the positivity of all ∈ minors from F (i0). This phenomenon has the following algebraic explanation. Let (1.23) F (u, v)= F (i) . i R(u,v) ∈[ This can be restated as 1 (1.24) F (u, v)= ∆ : i [1,r],u0 u, v0 v− , { u0ωi,v0ωi ∈ } 1 where u0 u stands for `(u)=`(u0)+`(u0− u) (the weak order on W ). Theorem 1.12. For any i R(u, v), the collection F (i) is a transcendence basis ∈ for the field of rational functions C(Gu,v). Furthermore, every minor in F (u, v) can be expressed as a ratio of two polynomials in the variables ∆ F (i) with nonnegative integer coefficients. ∈ Combining Theorems 1.2, 1.6, and 1.10 yields the following result. Theorem 1.13. For any i R(u, v), every minor in F (u, v) is a Laurent polyno- mial with integer coefficients∈ in the variables ∆ F (i). ∈ We suggest the following common refinement of Theorems 1.12 and 1.13. Conjecture 1.14. For any i R(u, v), every minor in F (u, v) is a Laurent poly- nomial in the variables ∆ F∈(i) with nonnegative integer coefficients. ∈ Note that Theorems 1.12 and 1.13 do not automatically imply Conjecture 1.14, since there do exist subtraction-free rational expressions that are Laurent polynomials although not with nonnegative coefficients (for example, think of (p3 + q3)/ (p + q)=p2 pq + q2). It is not hard− to derive the following special case of Conjecture 1.14 from [3, Theorem 3.7.4].

Theorem 1.15. Conjecture 1.14 holds for G = SLr+1 , when either u or v is the identity element e. 1.8. Fundamental determinantal identities. The subtraction-free rational expressions in Theorem 1.12 can be computed by an explicitly described algorithm. This algorithm is based on repeated application of the following generalized determinantal identities. The ﬁrst group of identities follows from [4, Corollary 6.6]. They correspond to pairs of simple roots that generate a root subsystem of type A2 or B2. There are

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similar identities for subsystems of type G2 (see [4, (4.8)–(4.11)]) although we will not reproduce them here. Theorem 1.16. Let u, v W and i, j [1,r]. (1) If a =a = 1 and∈`(vs s s )=∈`(v)+3,then ij ji − i j i

∆uωi,vsiωi ∆uωj ,vsj ωj =∆uωi,vωi ∆uωj ,vsisj ωj +∆uωi,vsj siωi ∆uωj ,vωj . (2) If a = 2, a = 1,and`(vs s s s )=`(v)+4,then ij − ji − i j i j ∆ ∆2 ∆ uωj ,vsisj ωj uωi,vsj siωi uωj ,vsj ωj =∆ ∆2 ∆ uωj ,vsj sisj ωj uωi,vsj siωi uωj ,vωj 2 +(∆uωi,vωi ∆uωj ,vsj sisj ωj +∆uωi,vsisj siωi ∆uωj ,vsj ωj ) and

∆uωi,vsiωi ∆uωi,vsj siωi ∆uωj ,vsj ωj =∆2 ∆ uωi,vsj siωi uωj ,vωj

+∆uωi,vωi (∆uωi,vωi ∆uωj ,vsj sisj ωj +∆uωi,vsisj siωi ∆uωj ,vsj ωj ) . (3) Each of the above identities has a companion identity, obtained by “trans- posing” all participating minors, i.e., by replacing every ∆γ,δ by ∆δ,γ . We also make use of the following new identity. Theorem 1.17. Suppose u, v W and i [1,r] are such that `(us )=`(u)+1 ∈ ∈ i and `(vsi)=`(v)+1.Then

(1.25) ∆ ∆ =∆ ∆ + ∆ aji . uωi,vωi usiωi,vsiωi usiωi,vωi uωi,vsiωi uω− j ,vωj j=i Y6 The proof of Theorem 1.17 is given in Section 2.3. For the type Ar , the identities of Theorems 1.16 and 1.17 become certain 3- term determinantal identities known since the early 19th century. We discuss their attribution in Section 4.6.

2. Preliminaries In what follows, we retain the notation and terminology introduced in Section 1.

2.1. Involutions. Following [4], we deﬁne involutive anti-automorphisms x xT (the “transpose”) and x xι of the group G by setting 7→ 7→ (2.1) aT = a (a H) ,x(t)T=x(t),x(t)T=x(t) ∈ i i i i and ι 1 ι ι (2.2) a = a− (a H) ,x(t)=x(t),x(t)=x(t). ∈ i i i i These two involutive anti-automorphisms commute with each other and with the 1 involutive anti-automorphism x x− of G. Hence these three maps generate 7→ the group isomorphic to (Z/2Z)3; in particular, any composition of them is again an involution. Notice that the involutions x xT and x xι preserve total 1 7→ ι 7→ nonnegativity, while x x− does not. Informally, x is a “totally nonnegative 1 7→ version” of x− . In the notation just introduced, the involution x xθ that was deﬁned by (1.11) is given by xθ =(xι)T =(xT)ι. 7→

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1 ι T The involutions x (x− ) and x x obviously preserve G0 = N HN,and we have 7→ 7→ − 1 ι T (2.3) [(x− ) ]0 =[x ]0 =[x]0 .

1 T ι 1 All three involutions x x− , x x and x x act on W by w w− . The relations between these7→ involutions7→ and the special7→ representatives introduced7→ in Section 1.4 are summarized in the following proposition. Proposition 2.1. We have

1 T 1 ι 1 ι 1 T (2.4) w − = w = w− = w , w− = w = w − = w . Proof. Since all three involutions are anti-automorphisms, it is enough to check (2.4) for w = si, in which case it follows by a calculation in SL2 . 2.2. Commutation relations. For convenience of exposition, we collect here some known commutation relations in G that will be used in our proofs. Recall

that xi(t)andxi(t) are deﬁned by (1.1), and α1,... ,αr are the simple roots of g. First of all, for every a H,wehave ∈ αi αi (2.5) axi(t)=xi(a t)a, axi(t)=xi(a− t)a.

The following relations between the elements xi(t) can be found, e.g., in [4, Section 3] (some of them appeared earlier in [19]). If aij = aji =0,then

(2.6) xi(t1)xj(t2)=xj(t2)xi(t1),

for any t1 and t2.Ifa = a = 1, then ij ji − t t t t (2.7) x (t )x (t )x (t )=x 2 3 x(t +t )x 1 2 i 1 j 2 i 3 j t +t i 1 3 j t +t 1 3 1 3 whenever t1 + t3 =0.Ifa = 2anda = 1, then 6 ij − ji − 2 1 1 2 1 1 (2.8) xi(t1)xj (t2)xi(t3)xj (t4)=xj(t2t3t4q− )xi(qp− )xj(p q− )xi(t1t2t3p− ) , where 2 2 p = t1t2 +(t1 +t3)t4,q=t1t2+(t1 +t3) t4 ; this relation holds whenever p =0andq= 0. In the case when a = 3,a = 6 6 ij − ji 1 (i.e., when αi and αj generate a root subsystem of type G2), there is also a −relation similar to (2.7) and (2.8). This relation is given in [4, (3.6)–(3.10)]; we will not reproduce it here. Each of the relations (2.6)–(2.8) has a counterpart for T the elements xi(t); it can be obtained by applying the anti-automorphism x x (cf. (2.1)). 7→ In conclusion, let us describe the commutation relations between the elements x (t)andx(t0). If i = j,then[e,f ]=0ing, hence i j 6 i j (2.9) xi(t)xj (t0)=xj(t0)xi(t),

for any t and t0. To handle the case i = j, we will need the following notation. For a nonzero t C and i [1,r], we denote ∈ ∈

hi t 0 (2.10) t = ϕi 1 , 0 t−

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h where ϕ : SL2 G is deﬁned by (1.7); alternatively, t i is an element of H i → uniquely determined by the condition that (thi )γ = tγ(hi) for any weight γ P . Then we have ∈ t0 t (2.11) x (t)x (t )=x (1 + tt )hi x i i 0 i 1+tt 0 i 1+tt 0 0 whenever 1 + tt0 = 0. This relation can be ﬁrst checked for SL2 by a simple matrix 6 calculation, and then extended to G by applying the homomorphism ϕi .Bythe same method, we verify the relations 1 1 h (2.12) x (t)x ( t− )=x(t− )t is i i − i i and

1 hi 1 xi(t)si = x (t− )t xi( t− ) , (2.13) i − 1 h 1 s x (t)=x( t− )t ix(t− ). i i i − i 2.3. Generalized determinantal identities. We start with some identities for the “principal minors” ∆ωi . The definition (1.6) implies that, for any x G , + ∈ x− N , x N,anda H,wehave ∈ − ∈ ∈ ω ω + ω ∆ i(x−x)=∆ i(xx )=∆ i(x), (2.14) ∆ωi(ax)=∆ωi(xa)=aωi∆ωi(x). In view of (2.3), we also have ω 1 ι ω T ω (2.15) ∆ i ((x− ) )=∆ i(x )=∆ i(x). The following property is less obvious. Proposition 2.2. For any x G, j = i,andt C, we have ∈ 6 ∈ ωi ωi ωi (2.16) ∆ (xxj (t)) = ∆ (xj (t)x)=∆ (x). Proof. It is possible to deduce the proposition from the commutation relations given in Section 2.2 but we prefer another proof based on representation theory. The group G acts by right translations in the space C[G] of regular functions on G.It is well known that every f C[G] generates a finite-dimensional subrepresentation ∈ of C[G]. In view of (2.14), the function ∆ωi is a highest weight vector of weight ωi ωi in C[G]. Since ωi(hj )=0forj=i, it follows that ∆ has weight 0 with 6 ωi respect to the subgroup ϕj(SL2)ofG(cf. (1.7)). Therefore, ∆ generates a trivial ωi ωi representation of ϕj (SL2). In particular, ∆ (xxj (t)) = ∆ (x), as desired. The ωi ωi equality ∆ (xj (t)x)=∆ (x) now follows from (2.15). Our next proposition justifies the validity of Definition 1.4. Proposition 2.3. For any x G and any j = i, we have ∈ 6 ωi ωi ωi (2.17) ∆ (xsj )=∆ (sjx)=∆ (x). Proof. Follows from (2.14), (2.16), and (1.8).

The extension of principal minors from the open subset G0 to the whole of G is given as follows. The Bruhat decomposition theorem implies that every x G can be written as ∈ + (2.18) x = x−awx , + for some x− N ,a H, w W,andx N; moreover, the elements a H and w W are∈ uniquely− ∈ determined∈ by x. ∈ ∈ ∈

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Proposition 2.4. If x G is expressed in the form (2.18),then ∈ aωi if wω = ω ; (2.19) ∆ωi(x)= i i 0 , otherwise . Proof. By (2.14), we have ω + ω ω ∆ i (x−awx )=a i∆ i(w). Thus, to prove (2.19) we only need to show that 1ifwω = ω ; (2.20) ∆ωi (w)= i i 0 , otherwise . The formula is obvious for w = e, the identity element of W . Hence we can assume that `(w) 1 and write w as us for some u W and j [1,r] with `(u)=`(w) 1. ≥ j ∈ ∈ − Since ∆ωi is a regular function on G,wehave

ωi ωi ωi (2.21) ∆ (w)=∆ (usj) = lim ∆ (uxj(t)sj ) . t 0 → Substituting into (2.21) the expression for xj (t)sj given by (2.13) and using (2.14), we obtain

ωi ωi(hj ) ωi 1 ∆ (w) = lim t ∆ (uxj (t− )) . t 0 → 1 1 Since `(u)=`(w) 1, the root u(αj) is positive, implying that uxj(t− )u− N . Again using (2.14),− we obtain ∈ − ∆ωi (w) = lim tδij ∆ωi (u) . t 0 → It follows that ∆ωi(u)ifj=i; ∆ωi (w)= 0, otherwise6 . This implies (2.20) by induction on `(w).

As a corollary, we obtain the following useful characterization of the set G0 . Corollary 2.5. An element x G has Gaussian decomposition if and only if ∈ ∆ωi (x) =0for any i [1,r]. 6 ∈ In subsequent proofs, we will also make use of the following identities.

Proposition 2.6. For any x =[x] [x]0[x]+ G0 and any w W , we have − ∈ ∈ ∆ωi,wωi (x) (2.22) ∆ωi,wωi ([x]+)= , ∆ωi (x)

∆wωi,ωi (x) (2.23) ∆wωi,ωi ([x] )= . − ∆ωi (x) Proof. Using (1.10) and (2.14), we obtain:

ωi ωi ωi ωi ∆ωi,wωi (x)=∆ (xw)=[x]0 ∆ ([x]+w)=∆ (x)∆ωi ,wωi ([x]+) , which proves (2.22). The proof of (2.23) is similar.

The transformation ( wo) permutes fundamental weights. We will use the no- − tation i i∗ for the induced permutation of the index set [1,r], so that 7→ (2.24) ω = wo(ω )(i [1,r]) . i∗ − i ∈

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Proposition 2.7. For any x G and any u, v W , we have ∈ ∈ T ι (2.25) ∆uωi,vωi (x)=∆vωi,uωi (x )=∆vwoω ,uwoω (x ) . i∗ i∗ Proof. Using (2.15) and (2.4), we obtain:

ωi 1 ωi 1 T ωi 1 T T ∆uωi,vωi (x)=∆ (u− xv)=∆ ((u − xv) )=∆ (v − x u)=∆vωi,uωi (x ) , which proves the ﬁrst equality in (2.25). To prove the second equality, let us introduce the anti-automorphism η of G by ι η(x)=woxwo.Sinceηpreserves H and interchanges N and N, it follows that η 1 − preserves G0 ,and[η(x)]0 = η([x]0)=wo[x]0− wo for any x G0 . Using (1.2) and ωi ω ∈ (2.24), we conclude that ∆ (η(x)) = ∆ i∗ (x). Hence

1 1 ωi ω ω ι 1 i∗ 1 i∗ − 1 − ∆uωi,vωi (x)=∆ (u− xv)=∆ (η(u− xv)) = ∆ (wo v x u− wo) ω ι ι i∗ 1 =∆ (wov xuwo)=∆vwoω ,uwoω (x ) , − i∗ i∗ as claimed.

1 1 Proof of Theorem 1.17. First of all, since siu− = si u− and vsi = v si, the deﬁ- nition (1.10) implies that it is enough to prove (1.25) in the case when u = v = e, the identity element. Thus, we only need to show that

a (2.26) ∆ ∆ ∆ ∆ = ∆− ji . ωi ,ωi siωi,siωi − siωi,ωi ωi,siωi ωj,ωj j=i Y6 As in the case of Proposition 2.2, our proof of (2.26) will rely on representation theory. Consider the representation ρ of the group G G in C[G]givenby × T ρ(x1,x2)f(x)=f(x1xx2) .

Let us denote the left- and right-hand sides of (2.26) by f1 and f2, respectively. We ﬁrst verify that the function f2 C[G] has the following properties: ∈ (1) f2 is a highest weight vector in the representation ρ, i.e., it is invariant under the subgroup N N G G; × ⊂ × γ (2) f2 has weight (γ,γ), where γ =2ω α;thatis,ρ(a1,a2)f2 =(a1a2) f2 for i− i any a1,a2 H; ∈ (3) f2(e)=1(hereestands for the identity element of G).

Property (3) is trivial, while (1) follows from (2.14). Also by (2.14), f2 has weight ( j=i ajiωj, j=i ajiωj). To prove (2), it is enough to show that − 6 − 6 j=i ajiωj = γ; but this follows from the equality − 6 P P

(2.27)P ajiωj = αi , j [1,r] ∈X which can be taken as a deﬁnition of the Cartan matrix. Properties (1)–(3) uniquely determine the restriction of f2 to G0 .SinceG0is dense in G,andf2 is regular, these properties uniquely determine f2 .Itremains to show that f1 satisﬁes (1)–(3). The normalization condition (3) follows from Proposition 2.4; indeed, in view of (2.19),

∆ωi,ωi (e)=∆siωi,siωi (e)=1, ∆siωi,ωi (e)=∆ωi,siωi (e)=0.

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To prove that f1 satisfies (2), notice that for any u, v W , the function ∆ ∈ uωi,vωi has weight (uωi,vωi) (this follows from (1.10) and (1.2)). Hence both summands in f1 have weight (ωi + siωi,ωi +siωi)=(γ,γ). To prove that f1 satisfies (1), we first notice that, in view of (2.25), we have T f1(x )=f1(x) for any x G, hence ρ(x1,x2)f1 = ρ(x2,x1)f1 for any x1,x2 G. ∈ ∈ Therefore, it suffices to show that f1 is invariant under the action of N by right translations. Let E1,... ,Er be the infinitesimal right translation operators on C[G] defined by d E (f)(x)= f(xx (t)) =0 ; j dt j |t each Ej is a derivation of the ring C[G]. It is enough to show that Ej f1 =0for all j.Ifj=i,thenEj annihilates all four minors that appear in f1 (this follows 6 1 from the fact that s x (t)s − N), hence E f1 = 0. It remains to prove that i j i ∈ j Eif1 = 0. Clearly, we have

(2.28) Ei∆ωi,ωi = Ei∆siωi,ωi =0. We claim that

(2.29) Ei ∆ωi ,siωi =∆ωi,ωi ,Ei∆siωi,siωi =∆siωi,ωi . Combining (2.28) and (2.29) and using the Leibniz rule, we obtain

E f1 =E (∆ ∆ ∆ ∆ ) i i ωi,ωi siωi,siωi − siωi,ωi ωi,siωi =∆ ∆ ∆ ∆ =0, ωi,ωi siωi,ωi − ωi,ωi siωi,ωi as required. We will deduce (2.29) from the following lemma which is a standard fact in the representation theory of SL2.

hi k Lemma 2.8. Suppose f C[G] is such that Eif =0and f(xt )=tf(x)for ∈ k some k 0.Letf0 C[G]be given by f 0(x)=f(xsi).ThenE(f0)=k!f. ≥ ∈ i

The ﬁrst equality in (2.29) follows by applying this lemma to f =∆ωi,ωi (in this

case, k =1andf0 =∆ωi,siωi ). Similarly, the second equality in (2.29) follows by

applying Lemma 2.8 to f =∆siωi,ωi (in this case, k =1andf0 =∆siωi,siωi ). This completes the proof of Theorem 1.17. 2.4. Affine coordinates in Schubert cells. For every w W , the corresponding Schubert cell (BwB)/B G/B is the image of the Bruhat∈ cell BwB under the natural projection of G onto⊂ the flag variety G/B. Let the subgroups N+(w) N and N (w) N be defined by ⊂ − ⊂ − 1 1 (2.30) N+(w)=N wN˜ w˜− ,N(w)=N w˜− Nw,˜ ∩ − − −∩ wherew ˜ is any representative of w in G;sinceHnormalizes N and N ,these subgroups do not depend on the choice ofw ˜. The following proposition is essentially− well known (cf. [11, Corollary 23.60]). Proposition 2.9. An element x G lies in the Bruhat cell BwB if and only if, ∈ 1 for some (equivalently, any) representative w˜ G of w, we have w˜− x G0 and 1 ∈ ∈ [˜w− x] N (w). Furthermore, the element − ∈ − 1 1 (2.31) y+ = π+(x)=w ˜[˜w− x] w˜− N+(w) − ∈ does not depend on the choice of w˜, and the correspondence π+ : x y+ induces a 7→ biregular isomorphism between the Schubert cell (BwB)/B and N+(w).

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Using the transpose map x xT , one obtains a counterpart of Proposition 2.9 for the opposite Bruhat cell B 7→wB . − − Proposition 2.10. An element x G lies in B wB if and only if, for some ∈ − − 1 1 (equivalently, any) representative w˜ G of w, we have xw˜− G0 and [xw˜− ]+ ∈ ∈ ∈ N+(w). Furthermore, the element 1 1 (2.32) y = π (x)=w ˜− [xw˜− ]+w˜ N (w) − − ∈ − does not depend on the choice of w˜, and the correspondence π : x y induces a biregular isomorphism between the “opposite Schubert cell” −B (7→B wB− ) and N (w). −\ − − − The group N (w) is a unipotent Lie group of dimension ` = `(w), hence it is − isomorphic to the affine space C` as an algebraic variety. We will associate with any i =(i1,... ,i`) R(w) the following system of affine coordinates on N (w). `∈ − For (p1 ,... ,p`) C ,weset ∈ 1 (2.33) yi(p1,... ,p`)=w− si x (p1) si x (p`) . · 1 i1 ··· ` i` Also, let us define

(2.34) wk = wk,i = si` si` 1 sik − ··· 1 for k [1,`+1],sothatw1 =w− and w +1 = e. ∈ ` Proposition 2.11. The map (p1,... ,p ) y = y (p1,... ,p ) is a biregular iso- ` 7→ i ` morphism between C` and N (w). The inverse map is given by − (2.35) p =∆ (y) . k wkωik,wk+1ωik Proof. We can rewrite (2.33) as ` y (p ,... ,p )= w x (p )w 1 . i 1 ` k+1 ik k k+1 − =1 kY Each factor w x (p )w 1 belongs to the root subgroup in G corresponding k+1 ik k k+1 −

to the root wk+1(αik ), and these are all the root subgroups in N (w)(cf.[5, VI, 1.6]). This− implies the ﬁrst statement in Proposition 2.11. To− prove (2.35), we set i0 =(i1,... ,ik 1)andi00 =(ik+1,... ,i`)sothati=(i0,ik,i00). Let y0 = − yi (p1,... ,pk 1)andy00 = yi (pk+1,... ,p`). In view of (2.33), we have 0 − 00 1 1 (2.36) w − yw = y (s x (p )) (w − y w ) . k k+1 0 ik ik k k+1 00 k+1

In this decomposition, the ﬁrst factor y0 belongs to N (si1 sik 1) N , while − ··· − ⊂ − 1 1 1 the last factor wk−+1y00wk+1 belongs to wk−+1 N (wk−+1) wk+1 N. Using (2.14) and (2.13), we conclude that − ⊂

ωi 1 ωi ∆ (y)=∆ k(w − yw +1)=∆ k(s x (p )) wkωik ,wk+1ωik k k ik ik k h ωi 1 ik 1 =∆ k(x ( p− )p xi (p− )) = pk , ik − k k k k as claimed. Note for future use that a similar argument allows us to prove that, for any i [1,r], k [1,`+ 1], and y N (w), we have ∈ ∈ ∈ −

(2.37) ∆wk ωi,wkωi (y)=1.

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This follows from a decomposition similar to (2.36):

1 1 wk − ywk = y0 (wk − y00wk) ,

1 1 1 where y0 N (si1 sik 1) N ,andwk − y00wk wk− N (wk− ) wk N. As a corollary∈ − of··· Proposition− ⊂ − 2.11, we obtain deﬁning∈ equations− for N⊂ (w)asa − subvariety in N . Notice that N = N (wo). Hence, for every j =(j1,... ,jn) − − − ∈ R(wo), any element y N can be uniquely written as y = yj(p1,... ,pn)for n∈ − some (p1,... ,pn) C (here n = `(wo)). Let us choose j so that its ﬁrst n ` ∈ 1 − indices form a reduced word j1 for wow− , while the last ` indices form a reduced

word j2 for w.Thenwrite(wo)k=sjnsjn 1 sjk for k [1,n+ 1], in agreement with (2.34). Finally, let us denote − ··· ∈

1 N 0 (w)=N w˜− N w,˜ − −∩ − wherew ˜ is any representative of w in G (cf. (2.30)). The following proposition is an immediate consequence of Proposition 2.11 and the deﬁnition (2.33).

Proposition 2.12. Every y N is uniquely written as y = y1y2 with y1 N 0 (w) ∈ − ∈ − and y2 N (w). In the above notation, if y = yj(p1,... ,pn),then ∈ − 1 y1=w− yj (p1,... ,pn `)w,y2=yj (pn `+1,... ,pn) . 1 − 2 − Hence y lies in N (w) if and only if −

(2.38) ∆( ) ( ) (y)=0 wo kωjk , wo k+1ωjk for k =1,... ,n `. − 2.5. y-coordinates in double Bruhat cells. Let us ﬁx a pair (u, v) W W u,v u,v ∈ × and consider the open subset G0 = G G0 consisting of the elements x in the double Bruhat cell Gu,v that admit Gaussian∩ decomposition (1.5). In view of u,v u,v Propositions 2.9 and 2.10, the restrictions π+ : G N+(u)andπ :G → u,v − → N (v) are well deﬁned. Let us also introduce the map π0 : G0 H by − → (2.39) y0 = π0(x)=[x]0 , thus obtaining the map

u,v u,v π =(π+,π0 ,π ):G0 N+(u) H N (v) . − → × × − u,v u,v For x G0 , we will write π (x)=(y+,y0 ,y ) and call this triple the y- coordinates∈ of x. −

Example 2.13. Let G = SL2(C), and let u = v = wo (cf. Example 1.8). A matrix x11 x12 u,v x = with determinant 1 belongs to G0 if and only if x11 =0, x21 x22 6 x12 =0,x21 = 0. Using formulas (2.31), (2.32), and (2.39), we see that the y-coordinates6 of6 x are given by

1 1 x11x21− x11 0 10 (2.40) y+ = ,y0= 1 ,y= 1 . 01 0 x− − xx− 1 11 11 12 Our use of the term “coordinates” for the triple (y+ ,y0 ,y ) is justiﬁed by the following statement. −

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Proposition 2.14. The map πu,v is a biregular isomorphism u,v 1 1 G0 (N+(u) G0u− ) H (N (v) v− G0) . → ∩ × × − ∩ The inverse isomorphism (y+,y0,y ) x is given by − 7→ (2.41) [x] =[y+u˜] , [x]0 =y0, [x]+ =[˜vy ]+ , − − − where u˜ and v˜ are arbitrary representatives of u and v.

Proof. By Proposition 2.9, any x BuB canbewrittenasx=y+ub˜,where ∈ y+=π+(x)andb B. It follows that x G0 if and only if y+u˜ G0, and if this ∈ ∈ ∈ is the case, then [x] =[y+u˜] . Similarly, an element x B vB lies in G0 if and − − ∈ − u,v− only ifvy ˜ G0,andthen[x]+ =[˜vy ]+. It follows that π is an embedding u,v − ∈ 1 − 1 of G0 into (N+(u) G0u− ) H (N (v) v− G0), and that the inverse map ∩ × × − ∩ is given by (2.41). The same argument shows that if the triple (y+ ,y0,y ) lies in 1 1 − (N+(u) G0u− ) H (N (v) v− G0), then the element x given by (2.41) lies u,v ∩ × × − ∩ in G0 , and we are done. The following proposition is immediate from the deﬁnitions.

1 1 T 1 u,v T v− ,u− Proposition 2.15. We have N+(u) = N (u− ) and (G0 ) = G0 .If u,v − T T T x G0 has y-coordinates (y+,y0,y ),thenx has y-coordinates (y ,y0,y+). ∈ − − This proposition shows that the transpose map “interchanges” the coordinates y+ and y , so that any statement about y has a counterpart for y+. For instance, Proposition− 2.9 is a counterpart of Proposition− 2.10 in this sense. u,v Proposition 2.16. Suppose x G0 has the y-coordinates (y+,y0,y ).Then ∈ − 1 1 1 1 1 1 (2.42) [u− x]0− =[y+u]0y0− , [xv− ]0− =y0− [vy ]0 . −

Proof. By Proposition 2.9, x = y+ub for some b B. It follows that y0 =[y+u]0[b]0. ∈ 1 1 On the other hand, u− x = u− y+ub N b, hence ∈ − 1 1 [u− x]0 =[b]0 =y0[y+u]0− . This proves the ﬁrst equality in (2.42); the second one follows by Proposition 2.15.

It will be of special importance for us to specialize Proposition 2.14 to the case when (u, v)=(e, w), where e is the identity element of W ,andw W is arbitrary. e,w e,w w ∈ Then we have G0 = G = HN where (2.43) N w = N B wB . ∩ − − Specializing Proposition 2.14 to this case, we obtain the following statement. Proposition 2.17. For any w W ,themapπ :B wB N (w) restricts to a w ∈ 1 − − − → − biregular isomorphism N N (w) w− G0. The inverse isomorphism N (w) 1 w → − ∩ − ∩ w− G0 N is given by y [˜wy]+,wherew˜is an arbitrary representative of w. → 7→ Using Proposition 2.15, we see that Proposition 2.14 is equivalent to its special case given by Proposition 2.17 combined with the following decomposition:

1 u,v u− T v (2.44) G0 =(N ) HN .

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2.6. Factorization problem in Schubert cells. In this section we recall some results from [4] concerning a version of the factorization problem for the variety N w = N B wB (cf. (2.43)). We will need the following analogue of Theorem 1.2 which is essentially∩ − − due to G. Lusztig [18] (cf. also [4, Proposition 1.1]).

Proposition 2.18. For any w W and any reduced word i =(i1,... ,i`) R(w), ∈ ∈ ` the map (t1,... ,t`) xi1 (t1) xi`(t`) is a biregular isomorphism between C=0 and a Zariski open subset7→ of N w···. 6 We will give explicit formulas for the inverse of the product map in Proposi- tion 2.18.

w Theorem 2.19. Let i =(i1,... ,i ) R(w), and let x = x (t1) x (t ) N ` ∈ i1 ··· i` ` ∈ with all tk nonzero complex numbers. Then the tk are recovered from x by 1 aj,ik (2.45) tk = ∆wk+1ωj,ωj (y)− , ∆w ω ,ω (y)∆w ω ,ω (y) k ik ik k+1 ik ik j=i Y6 k where wk is given by (2.34),andy=π (x) N (w)(cf. (2.32)). − ∈ − This theorem is a reformulation of [4, Theorems 1.4, 6.2]. Here we present a new proof which is in some sense more elementary than the one in [4], and also provides additional information that we will need later on. Proof. There is nothing to prove if w = e, so we will assume that `(w)=` 1. ≥ Let y = π (x)andz=wy. By Proposition 2.17, z G0 and x =[z]+.Letus − ∈ write i1 = i, and denote w0 = s w, i0 =(i2,... ,i ) R(w0), x0 = x ( t1)x = i ` ∈ i − w0 xi2 (t2) xi`(t`) N , y0 = π (x0) N (w0), and z0 = w0y0.Hereisthekey lemma.··· ∈ − ∈ −

Lemma 2.20. In the notation just introduced, let us write y0 = yi0 (p2,... ,p`),in accordance with Proposition 2.11. Then y = yi(p1,p2,... ,p`),wherep1 is given by

αi 1 1 (2.46) p1 =∆siωi,ωi (xi([z0]0− t1− )[z0]− ) . − Furthermore, we have

ωi αi ωi (2.47) t1 =[z0]0 − [z]0− .

Proof. Let us temporarily denotey ˜ = yi(p1,p2,... ,p`)and˜z=wy˜,wherep1 is given by (2.46); our goal is to show thaty ˜ = y andz ˜ = z. By Proposition 2.17, it 1 suﬃces to show that [˜z]+ = x,orequivalentlythat˜zx− B . ∈ − By Proposition 2.12 (applied to w = si), formula (2.46) implies that

αi 1 1 1 (2.48) xi([z0]0− t1− )[z0]− = siy00si − xi(p1) , − where y00 N . Using (2.5) and (2.12), we can rewrite the left-hand side of (2.48) as follows:∈ −

αi 1 1 1 1 1 1 1 xi([z0]0− t1− )[z0]− =[z0]0xi(t1− )[z0]0− [z0]− =[z0]0xi(t1− )x0(z0)− − − 1 1 hi 1 1 =[z0]0x(t− )x( t1)x(z0)− =[z0]0st− x( t− )x(z0)− . i 1 i − i 1 i − 1 1 Substituting this expression into (2.48) and using the fact thatz ˜ = si− xi(p1)z0, we can rewrite (2.48) as follows:

1 1 1 hi 1 (2.49) zx˜ − =(y00)− s − [z0]0s t− x ( t− ) . i i 1 i − 1

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1 It follows thatzx ˜ − B , hencey ˜ = y andz ˜ = z. Since the left-hand side of ∈ − (2.49) is equal to [z] [z]0 , it follows that − 1 hi (2.50) [z]0 = si − [z0]0si t1− . Finally, (2.47) follows from (2.50) by applying the character a aωi to both sides and using (1.2). 7→

Note that (2.46) can be simpliﬁed as follows: 1 (2.51) p =∆ 1 (xy − )/∆ 1 (x); 1 ωi,w0− ωi 0 ωi,w− ωi since we will not need this formula, the proof is left to the reader. Continuing with the proof of Theorem 2.19, let us deﬁne, for k =1,...,`:

1 (k) w− (k) (k) (k) 1 (k) x =xi (tk) xi (t`) N k ,y =π (x ),z =w− y . k ··· ` ∈ − k Applying (2.47) with x replaced by x(k) yields ω α ω (k+1) ik − ik (k) − ik (2.52) tk =[z ]0 [z ]0 . On the other hand, combining the deﬁnition (1.10) with (2.27), we can rewrite (2.45) as follows:

ω α ω 1 ik − ik 1 − ik (2.53) tk =[wk−+1y]0 [wk− y]0 . Comparing (2.53) with (2.52), we see that Theorem 2.19 would follow from the 1 (k) 1 equality [wk− y]0 =[z ]0. The latter is obtained by observing that wk− y = 1 1 − (k) wk− w z =˜yz ,where˜y=y(i1,...,ik 1)(p1,...,pk 1) N (thisy ˜ was denoted − − ∈ − by y0 in (2.36)). 2.7. Totally positive bases for N (w). Although most of the results in this section were obtained in [4], we prefer− to give independent proofs here; in some cases, this will allow us to refine the statements in [4]. We start with the following general definition. Definition 2.21. Let F be a finite collection of functions on a set X. A subset Fis called a totally positive base for F if is a minimal (with respect to B⊂inclusion) subset of F with the property that everyB f F is a subtraction-free expression (i.e., a ratio of two polynomials with nonnegative∈ integer coefficients) in the elements of . B For every w W , let us denote ∈ 1 (2.54) F (w)= ∆ (y):i [1,r],w0 w00 w− . { w00 ωi,w0ωi ∈ } 1 (As earlier in (1.23), w0 w00 stands for `(w00)=`(w0)+`(w0− w00).) To every reduced word i =(i1,... ,im) R(w) we associate three collections of regular functions on the group N (w): ∈ − F1(i)= ∆wω,ω :1 k m , { kik ik ≤ ≤ } (2.55) F2(i)= ∆w1ω,w ω :2 k m+1 , { − ik k ik ≤ ≤ } F(i)= ∆ : i [1,r], 1 k l m+1 , {wkωi,wlωi ∈ ≤ ≤ ≤ } where w = s s (cf. (2.34)). k im ··· ik

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Theorem 2.22. For any reduced word i =(i1,... ,i ) R(w), each of the col- m ∈ lections F1(i) and F2(i) of regular functions on N (w) is a transcendence basis for − C(N (w)) and a totally positive base for F (w). − Proof. Let us ﬁrst deal with F1(i). The most important part of the proof is to show that every minor in F (w) is a subtraction-free expression in the minors from F1(i). Since we obviously have F (w)= F(i), i R(w) ∈[ this statement will directly follow from Lemmas 2.23 and 2.24 below.

Lemma 2.23. For any two reduced words i, i0 R(w), every minor in F1(i0) is a ∈ subtraction-free expression in the minors from F1(i). Proof. This is an immediate corollary of [4, Corollary 6.7]. The proof in [4] is based on repeated applications of determinantal identities of Theorem 1.16.

Lemma 2.24. Every minor in F (i) is a subtraction-free expression in the minors from F1(i).

Proof. We need to show that every minor ∆wk ωi,wlωi ,fori [1,r]and1 k l m+ 1, is a subtraction-free expression in the minors ∆ ∈ . Recall that,≤ by≤ ≤ wkωi,ωi convention, wm+1 = e, so the statement is trivial for l = m + 1. By (2.37), it also holds for k = l, since the corresponding minor equals 1. Thus we may assume that 1 kl,orl0 =l, k0 >k. Using induction with respect to this linear order, it is enough to show that for every (k, l) such that 1 k

∆w ωj,w ωj with j [1,r]and(k0,l0)<(k, l). The latter follows from the identity k0 l0 ∈ (1.25) applied to u = wk+1 and v = wl+1 . Indeed, this identity can be rewritten as a − ji ∆wkωi,wl+1ωi ∆wk+1ωi,wlωi + j=i ∆wk+1ωj ,wl+1ωj ∆ = 6 , wkωi,wlωi ∆ wk+1ωi,wl+1Qωi providing a desired subtraction-free expression.

Lemma 2.24 implies in particular that each minor ∆ (y) is a rational wkωik ,wk+1ωik function of the minors from F1(i). By Proposition 2.11, it follows that F1(i)isa transcendence basis for C(N (w)), hence it is a totally positive base for F (w). − To prove that F2(i) has the same properties, we will apply the anti-automorphism τw of G given by 1 θ 1 (2.56) τw(y)=w(y− ) w − ,

where θ was deﬁned in (1.11). In view of (2.4), if τw(y)=y0,then

T T (2.57) y=τ 1(y),τ(y)=y . w− 0 w 0 A straightforward check shows that 1 1 (2.58) τw 1 (N+(w)) = N+(w− ) ,τw(N(w)) = N (w− ) . − − −

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Lemma 2.25. Let y G, and let y0 = τw(y) for some w W . For any w0,w00 W 1 ∈ 1 ∈ ∈ such that w0 w− and w00 w− , we have

(2.59) ∆w00ωi,w0ωi (y)=∆ww0ωi,ww00ωi (y0) . 1 1 Proof. In view of (1.9) and (2.4), the conditions w0 w− and w00 w− imply that 1 1 − 1 (ww0) − = w0 w − , ww00 = w w00 . Combining this with (2.15), we obtain: (2.60) 1 ωi 1 ωi − 1 ∆ww0ωi,ww00 ωi (y0)=∆ ((ww0) − y0ww00)=∆ (w0 w− y0 ww00) 1 ωi − 1 θ ωi 1 =∆ (w0 (y− ) w00)=∆ (w00 − yw0)=∆w00ωi,w0ωi (y) , as claimed.

1 By Lemma 2.25, the anti-automorphism τw transforms F (w)intoF(w− ), and 1 F2(i)intoF1(i∗), where i∗ =(i ,... ,i1) R(w− )isiwritten backwards. Thus m ∈ the fact that F2(i) is a transcendence basis for C(N (w)) and a totally positive − base for F (w), follows from the same properties for F1(i) that we already proved. This completes the proof of Theorem 2.22.

2.8. Total positivity in y-coordinates. Let N 0 N denote the multiplicative ≥ ⊂ semigroup generated by the elements xi(t)fori [1,r]andt>0. For every w W , let us denote (cf. [4]) ∈ ∈ w w (2.61) N>0 = N 0 N = N 0 B wB . ≥ ∩ ≥ ∩ − − The following analogue of Theorem 1.3 is due to G. Lusztig [18] (cf. Proposi- tion 2.18).

Proposition 2.26. For any w W and any reduced word i =(i1,... ,i`) R(w), ∈ ` w∈ the map (t1,... ,t`) xi1 (t1) xi`(t`) restricts to a bijection R 0 N 0. 7→ ··· > → > We will use Theorem 2.22 to obtain the following criteria for total positivity.

w Theorem 2.27. Let x N , let y = π (x) N (w), and let i =(i1,... ,i`) R(w). Then the following∈ conditions are− equivalent:∈ − ∈ w (1) x N>0 ; (2) ∆(∈y) > 0 for any ∆ F (w); ∈ (3) ∆(y) > 0 for any ∆ F1(i); ∈ (4) ∆(y) > 0 for any ∆ F2(i). ∈ Proof. The equivalence of (2), (3) and (4) is immediate from Theorem 2.22. Let us show the equivalence of (1) and (3). By Proposition 2.26, every x N w is ∈ >0 of the form x = x (t1) x (t )forsomet1,... ,t > 0. By Theorem 2.19, each i1 ··· i` ` ` t is a monomial in ` variables ∆(y):∆ F1(i) . It follows that the monomial k { ∈ } transformation from ∆(y):∆ F1(i) to t1,... ,t` is invertible (an explicit expression for the inverse{ transformation∈ } was{ given in [4,} Theorem 4.3] but we will not need it here). Thus every ∆(y)with∆ F1(i) is a Laurent monomial in ∈ t1,... ,t . Hence ∆(y) > 0, and (1) (3) is proved. ` ⇒ To prove (3) (1), suppose that ∆(y) > 0for∆ F1(i). Let us deﬁne t1,... ,t` ⇒ w ∈ via (2.45), and letx ˜ = xi1 (t1) xi`(t`) N>0. Settingy ˜ = π (˜x), we see that ··· ∈ −

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∆(y)=∆(˜y) for any ∆ F1(i). By Lemma 2.24, we have ∆(y)=∆(˜y) for any ∈ ∆ F (i). In particular, ∆w ω ,w ω (y)=∆w ω ,w ω (˜y)fork=1,... ,`. ∈ k ik k+1 ik k ik k+1 ik Using (2.35), we conclude that y =˜yand so x =˜x Nw ,proving(3) (1). ∈ >0 ⇒ 1 Corollary 2.28. The map τw : N (w) N (w− ) (cf. (2.58)) restricts to a 1 − − w w− → bijection π (N>0) π (N>0 ). − → − Proof. We have already observed that by Lemma 2.25, τw transforms F (w)into 1 F(w− ). The corollary then follows from Theorem 2.27. We will now show that using y-coordinates (i.e., passing from a double Bruhat u,v u,v cell G to the open subset G0 ) will not create problems in the study of totally positive varieties. The following proposition is due to G. Lusztig [18]; for the convenience of the reader, we provide a proof.

T Proposition 2.29. We have N 0 = N G 0, N 0 = N G 0 and G 0 = T ≥ ∩ ≥ ≥ − ∩ ≥ ≥ N 0H>0N 0. In particular, G 0 G0 , i.e., any totally nonnegative element in G≥admits the≥ Gaussian decomposition.≥ ⊂ Furthermore, for any u, v W , the totally u,v ∈ positive variety G>0 decomposes as

1 u,v u− T v (2.62) G>0 =(N>0 ) H>0N>0 , in the notation of (2.61).

Proof. By the deﬁnition of G 0, every totally nonnegative element x G has the ≥ ∈ form (cf. (1.3)) x = xi(a; t1,... ,tm), where i =(i1,... ,im) is some word in the alphabet [1,r] [1,r], the tk are positive real numbers, and a H>0.Wesay that i is unmixed∪ if all the indices from [1, r] precede those from [1∈,r]. By repeated application of the commutation relations (2.5), (2.9) and (2.11), we can transform x

to the form x = xi0 (a0; t10 ,... ,tm0 ) for an unmixed word i0, a0 H>0,andalltk0 >0. T ∈ This proves the decomposition G 0 = N 0H>0N 0. The equalities N 0 = N G 0 ≥ ≥ ≥ ≥ T ≥ ∩ and N 0 = N G 0 follow from this decomposition of G 0 and the uniqueness of the≥ Gaussian− ∩ decomposition.≥ Finally, (2.62) is proved by≥ the same argument combined with (2.44).

Combining Propositions 2.29 and 2.15 with Theorem 2.27, we obtain the follow- u,v ing description of the totally positive variety G>0 in terms of y-coordinates. Theorem 2.30. An element x Gu,v lies in Gu,v if and only if its y-coordinates ∈ 0 >0 (y+,y0,y ) satisfy the following properties: − ∆(y ) > 0 for any ∆ F (v); • −T ∈ 1 ∆(y ) > 0 for any ∆ F (u− ); • + ∈ y0 H 0. • ∈ > 3. Proofs of the main results This section contains proofs of the main results in Section 1.

3.1. Proofs of Theorems 1.1, 1.2, and 1.3.

Proof of Theorem 1.1. We will explicitly construct a desired biregular isomorphism + ( )+ ( ) of Gu,v with a Zariski open subset of Cr ` u ` v with the help of a “twisted”

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version of y-coordinates (cf. Section 2.5). We ﬁx a representativeu ˜ of u,and u,v associate to any x G atriple(y(+),y(0),y( ))givenby ∈ − 1 1 (3.1) y(+) = π+(x),y( )=π (x− ),y(0) =[˜u− x]0 ; − − in view of Propositions 2.9 and 2.10, this triple is well deﬁned and belongs to 1 N+(u) H N (v− ). Our statement is a consequence of the following. × × − Proposition 3.1. The correspondence x (y(+),y(0),y( )) given by (3.1) is a u,v 7→ − 1 biregular isomorphism of G with the Zariski open subset of N+(u) H N (v− ) 1 × × − consisting of triples such that y( )y(+) vG0u− . − ∈ Proof. The proof is essentially the same as that of Proposition 2.14; the inverse of the correspondence (3.1) is given by 1 1 x = y(+)u˜[˜v− y( )y(+)u˜]+− y(0) , − wherev ˜ is any representative of v. Proof of Theorem 1.2. By Theorem 1.1, every double cell Gu,v is smooth, and dim(Gu,v)=dim(H Cm). It follows from the Grothendieck-Zariski factorization theorem [14, Theorem× 8.12.6] that if X and Y are irreducible algebraic varieties over C of the same dimension, and Y is smooth, then any injective morphism from X to Y is an open embedding. Therefore, Theorem 1.2 is implied by the following proposition. (We thank Michel Brion for explaining this implication to us and providing the reference [14].)

Proposition 3.2. For every u, v W and i =(i1,... ,im) R(u, v),themapxi ∈ m u,v ∈ restricts to an injective regular map H C=0 G . × 6 → m u,v m Proof. First let us show that xi(H C=0) G . We will show that xi(H C=0) m× 6 ⊂ × 6 ⊂ B vB ; the inclusion xi(H C=0) BuB is proved similarly (or deduced from − − × 6 ⊂ the previous one with the help of the transpose map). Let

k1 <

B w0B B w00B = B w0w00B − − · − − − − whenever `(w0w00)=`(w0)+`(w00) (cf. [5, IV.2.4]). m u,v It remains to show that the map xi : H C=0 G is injective. There is × 6 → nothingtoproveifu=v=e, so we can assume that m = `(u)+`(v) 1. Suppose that i = i [1,r] (the case when i [1, r] is treated in the same way).≥ Denote m ∈ m ∈ v0 = vsi so that i0 =(i1,... ,im 1) R(u, v0). Now suppose − ∈ xi(a; t1,... ,tm)=xi(a0;t10,... ,tm0 ) , m where (a; t1,... ,tm)and(a0;t10,... ,tm0 )belongtoH C=0. Multiplying both sides × 6 of the last equality on the right by x ( t0 ), we obtain: i − m

(3.2) xi(a; t1,... ,tm 1,tm tm0 )=xi0(a0;t10,... ,tm0 1) . − − −

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u,v If t = t0 , then the left-hand side of (3.2) lies in G , while the right-hand side lies m 6 m in Gu,v0 . But this is impossible because the double Bruhat cells are disjoint. Thus

tm = tm0 , and the desired injectivity follows by induction on m. This completes the proof of Proposition 3.2 and Theorem 1.2. Proof of Theorem 1.3. If a double reduced word i R(u, v) is unmixed, i.e., all the indices from [1, r] precede those from [1,r], then∈ our statement follows by combining (2.62) with Proposition 2.26. The statement for an arbitrary i R(u, v) can be reduced to the case of an unmixed i by the argument used in the∈ proof of Proposition 2.29, i.e., by repeated application of the commutation relations (2.5), (2.9) and (2.11). 3.2. Proofs of Theorems 1.6 and 1.7. Proof of Theorem 1.6. The fact that the right-hand side of (1.12) is well deﬁned u,v for any x G follows from Propositions 2.9 and 2.10. Let us show that x0 = 1 1 u,v ∈u− ,v− ζ (x) G . Using (2.31), we can rewrite x0 as ∈ θ 1 1 1 1 x0 = u− y+− [xv− ] [xv− ]0 , − where y+ = π+(x). It follows that ux0 G0,and ∈ 1 θ (3.3) [ux0] =(y+− ) . − θ 1 Hence [ux0] N+(u) = N (u− ), and we conclude from Proposition 2.9 that 1 − ∈ − 1 x0 Bu− B. The inclusion x0 B v− B is proved in a similar way (or by using the∈ transpose map); the counterpart∈ − of (3.3)− is given by 1 θ (3.4) [x0v]+ =(y− ) , − where y = π (x) (cf. (2.32)). − − 1 1 u− ,v− We have proved that x0 G . To complete the proof of Theorem 1.6, it 1 ∈1 suﬃces to show that ζu− ,v− (ζu,v(x)) = x for any x Gu,v. Notice that (3.3) and (3.4) can be rewritten as ∈ θ θ 1 1 1 1 1 1 (3.5) [ux0] = u [u− x]− u− , [x0v]+ = v− [xv− ]+− v. − − 1 1 u− ,v− The desired equality ζ (x0)=xfollows by substituting these expressions into 1 1 u− ,v− the expression for ζ (x0) obtained from (1.12). The following proposition shows that the twist map respects the Gaussian decomposition.

u,v u,v Proposition 3.3. The twist map ζ : x x0 sends the open subset G0 to 1 1 u− ,v− 7→ G0 , and we have

1 1 1 1 (3.6) [x0]0 =[u− x]0− [x]0[xv− ]0− .

Proof. To show that x and x0 belong to G0 simultaneously, it suﬃces to rewrite (1.12) as θ 1 1 1 1 1 x0 = [u− x]0[u− x]+ x− [xv− ] [xv− ]0 . − In view of (2.3), this also implies (3.6).

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Let us now describe the twist map in terms of y-coordinates. Recall the deﬁnition (2.56) of the anti-automorphism τw of the group G. u,v Proposition 3.4. Suppose x G0 has y-coordinates (y+,y0,y ). Then the y- ∈ u,v − coordinates (y+0 ,y0,y0 ) of x0 = ζ (x) are given by −

y = τ 1 (y ) , +0 u− + 1 (3.7) y0 =[y+u]0y0− [vy ]0 , − y0 = τv(y ) . − −

Proof. The desired expressions for y+0 and y0 follow from (3.5); the expression for − y0 follows from (3.6) combined with (2.42).

u,v u,v Proof of Theorem 1.7. Let x G>0 , and let x0 = ζ (x). By Proposition 2.29, u,v ∈ x G0 so x has well-defined y-coordinates (y+,y0,y ). By Proposition 3.4, ∈ − the y-coordinates (y+0 ,y0,y0 )ofx0 are given by (3.7). By Theorem 2.30, the − triple (y+,y0,y ) satisfies the properties given there, and it suffices to check that − 1 1 (y+0 ,y0,y0 ) satisfies the same properties with (u, v) replaced by (u− ,v− ). In − view of (3.7) and (2.59), if ∆(y ) > 0 for any ∆ F (v), then ∆(y0 ) > 0for 1 − ∈ T − any ∆ F (v− ). Similarly, using (2.57) we obtain that if ∆(y+) > 0 for any ∈ 1 T ∆ F (u− ), then ∆(y+0 ) > 0 for any ∆ F (u). ∈ ∈ ωi It remains to show that y0 H>0. Applying the character a a to both sides ∈1 → of the equality y0 =[y+u]0 y0− [vy ]0 in (3.7), we obtain − ωi T ωi 1 (y0) =∆uωi,ωi (y+) y0− ∆v ωi,ωi (y ) . − −

1 ωi Since ∆ 1 F (v)and∆uω ,ω F (u− ), it follows that (y0 ) > 0 for any v− ωi,ωi ∈ i i ∈ 0 i [1,r]. Therefore, y0 H 0, as desired. ∈ 0 ∈ > 3.3. Proof of Theorem 1.9. First notice that the equivalence of (1.19) and (1.20) follows by applying the character a aωi to both sides of (1.20) and simplifying the result. In proving (1.18) and (1.20),7→ we will follow the same strategy that was used in the proof of Theorem 1.3: ﬁrst treat the case when i is unmixed, and then extend the result to the general case with the help of commutation relations (2.5), (2.9) and (2.11). Letusﬁrstassumethati R(u, v) is unmixed, i.e., all the indices from [1, r] precede those from [1,r]. Repeatedly∈ using (2.5), we conclude that in this case u,v x = xi(a; t1,...,tm) G0 , and the components in the Gaussian decomposition of x are given by ∈

α α i i`(u) [x] = xi (a− | 1| t1) xi (a− | |t`(u)) , (3.8) − 1 ··· `(u) [x]0 =a, [x]+ =x (t ) x (t ) . i`(u)+1 `(u)+1 ··· im m 1 1 u− ,v− Since by Theorem 1.6, x = ζ (x0), formula (3.6) implies that 1 1 (3.9) a =[x]0 =[ux0]0− [x0]0[x0v]0− . This proves (1.20) since a simple inspection shows that the right-hand side of (3.9) is equal to that of (1.20) when i is unmixed. Turning to the proof of (1.18), let us ﬁrst consider the case `(u)

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By (2.45), we have (3.10) 1 aj,i t = ∆ 1 (y ) k . k v− v

1 ∆ 1 (y )=∆ (y )=∆ (v y ) v− v

=∆ωj,v

Substituting the expressions given by (3.11) into (3.10), we express tk as a Laurent monomial in the minors ∆l(x0) given by (1.16). Using the notation from Section 1.6, this monomial can be written as follows: 1 a i ,i aj,i (3.12) tk = ∆l(x0)− | l| k ∆m+j(x0) k , ∆k(x0)∆k+ (x0) l:l

α i (ε(il) ε(il ))a i , i /2 a | k| = ∆l(x0) − − | l| | k | . 1 l m+r ≤ Y≤ Thus in order to check (1.18), it suﬃces to show that, for i unmixed and 1 k `(u), the right-hand side of (3.13) is equal to ≤ ≤ m+r (ε(i )/2+χ(k,l− ) ε(il)/2 χ(k,l)) a i , i ∆l(x0) l− − − | l| | k | ; =1 Yl this is again checked by direct inspection. Now let us prove (1.18) and (1.20) for an arbitrary double reduced word for u and v. Every such word can be obtained from an unmixed one by a sequence of mixed moves of the form ji ij . ··· ··· ··· ··· It therefore suﬃces to prove the following statement.

Lemma 3.5. Suppose a reduced word i0 R(u, v) is obtained from i R(u, v) by ∈ ∈ a mixed move. If (1.18) and (1.20) hold for i, then they also hold for i0.

Proof. Suppose i0 is obtained from i by interchanging ik = j and ik+1 = i. By (2.5), (2.9) and (2.11), the factorization parameters that appear in two factorizations

x = xi(a; t1,... ,tm)=xi0(a0;t10,... ,tm0 )

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of an element x Gu,v are related as follows. If i = j,then ∈ 6 (3.14) a0 =a, t0 =t (l/ k, k +1 ),t0 =t +1,t0 = t . l l ∈{ } k k k+1 k For i = j , a straightforward calculation using (2.5) and (2.11) shows that

ωp ωp δi,p (3.15) (a0) = a (1 + tktk+1)− ,

ε(il)ai, i (3.16) tl0 = tl(1 + tktk+1) | l| (l

1 (3.17) tk0 = tk+1(1 + tktk+1),tk0+1 = tk(1 + tktk+1)− ,

(3.18) tl0 = tl (l>k+1).

ωp We need to show the following: if we substitute the parameters tl and a given ωp by (1.18) and (1.19) into (3.14)–(3.18), then the resulting tl0 and (a0) satisfy the same formulas (1.18) and (1.19) with i replaced by i0. This is immediate from the deﬁnitions when i = j, so let us assume i = j. By the deﬁnition (1.16), we have 6 ∆l,i =∆l,i0 for l = k + 1, so we will denote this minor simply by ∆l .Thekey calculation is now6 as follows.

Lemma 3.6. In the above notation, if tk and tk+1 satisfy (1.18),then

∆k+1,i(x0)∆k+1,i(x0) (3.19) 1+tktk+1 = . ∆k(x0)∆(k+1)+ (x0)

Proof. Let us denote u0 = u k+2 and v0 = v

∆k+1,i =∆uω ,v ω , ∆k+1,i =∆us ω ,v s ω , (3.20) 0 i 0 i 0 0 i i 0 i i + ∆k =∆u0siωi,v0ωi , ∆(k+1) =∆u0ωi,v0siωi , so (3.19) takes the form

∆u0ωi,v0ωi (x0)∆u0 siωi,v0siωi (x0) (3.21) 1+tktk+1 = . ∆u0siωi,v0ωi (x0)∆u0ωi,v0siωi (x0)

On the other hand, if tk and tk+1 are given by (1.18), then m+r (χ(k,l− )+χ(k+1,l−) χ(k,l) χ(k+1,l)) a i ,i tktk+1 = ∆l(x0) − − | l| . =1 Yl By (1.17), we have 1ifl>k+1; (3.22) χ(k, l)+χ(k+1,l)= 1/2ifl k, k +1 ;  ∈{ }  0ifl

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If we substitute the expressions given by (1.18), (1.19) and (3.19) into the formulas (3.14)–(3.18), then an easy simpliﬁcation shows that they will be given by (1.18) and (1.19) for i0. This completes the proofs of Lemma 3.5 and Theorem 1.9.

3.4. Proofs of Theorems 1.11 and 1.12. We start by recalling a well-known property of reduced words in Coxeter groups (cf. [5, 15]). To state it, we will need the following notion. We call a d-move the transformation of a reduced word that replaces d consecutive entries i,j,i,j,... by j,i,j,i,...,forsomeiand j such that d is the order of sisj .Notethat,forgiveniand j, the value of d can be determined from the Cartan matrix as follows: if aij aji =0(resp.1,2,3), then d =2(resp.3,4,6). Proposition 3.7. Every two reduced words for the same element of a Coxeter group can be obtained from each other by a sequence of d-moves. Applying this proposition to the group W W , we conclude that every two × double reduced words i, i0 R(u, v) can be obtained from each other by a sequence of the following operations:∈ d-moves for each of the alphabets [1,r]and[1,r], and also mixed moves (cf. Section 3.3) and their inverses.

Proof of Theorem 1.12. Let us first prove that F (i) is a transcendence basis for the field C(Gu,v). By Theorem 1.1, F (i) is of cardinality dim Gu,v. It is therefore enough to show that F (i) generates C(Gu,v). In view of Theorem 1.6, it suffices to u,v show that the collection of “twisted” minors ∆k,i(x0) (cf. (1.16)) generates C(G ). u,v By Theorem 1.2, the field C(G ) is generated by the factorization parameters tk and aωi , and the claim follows by Theorem 1.9. The second statement of the theorem is a consequence of the following lemma.

Lemma 3.8. Suppose a double reduced word i0 is obtained from i by a d-move in one of the alphabets [1,r] and [1, r], or by a mixed move, or by the inverse of a mixed move. Then each element of the set diﬀerence F (i0) F (i) is a subtraction- free expression in the elements of F (i). \ Proof. For d-moves in [1,r]or[1,r], the desired subtraction-free expressions can be obtained from generalized Pl¨ucker relations in Theorem 1.16 (including the omitted relations of type G2); this part of the argument is essentially borrowed from [4, Proposition 6.10]. For mixed moves and their inverses, the statement follows in the same way from Theorem 1.17 (cf. the proof of Lemma 3.5 above). Lemma 3.8 and Theorem 1.12 are proved.

Proof of Theorem 1.11. It will suffice to show that the following are equivalent: u,v (1) x G>0 ; (2) ∆(∈x) > 0 for any ∆ F (u, v); (3) ∆(x) > 0 for any ∆ ∈ F (i). The equivalence of (2) and∈ (3) follows from Theorem 1.12. Let us show that 1 1 u,v u− ,v− (1) (3). By Theorem 1.7, if x G ,thenx0 G . The condition ⇒ ∈ >0 ∈ >0 (3) now follows by applying Theorems 1.3 and 1.10 to x0 and the reduced word 1 1 i∗ R(u− ,v− ) opposite to i. ∈It remains to show that (2) (1). First of all, by Corollary 2.5, (2) implies that u,v ⇒ x G ; moreover, [x]0 H 0 . By Proposition 2.15 and Theorems 2.30 and 2.27, ∈ 0 ∈ > is suffices to show that y = π (x)satisfies∆v 1ω,v ω (y ) > 0, for i [1,r]and − − − i 0 i − ∈

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1 any v0 v− . Using (2.41) and (2.22), we obtain (cf. (3.11)):

∆ωi,v0ωi (x) (3.23) ∆ 1 (y )= . v− ωi,v0ωi − ∆ 1 (x) ωi,v− ωi By (2), the right-hand side of (3.23) is positive, and the proof is complete.

4. GLn theory

Throughout this section, G = GLn(C) is the group of invertible n n matrices with complex entries. In this case, the problems under consideration become× quite natural questions in “classical” linear algebra, so we will formulate them here— and state our main results—in an elementary and self-contained way. We will not give any proofs though, since these results can be easily derived from the type A specializations of the corresponding statements in Section 1; pointers to these statements are provided, wherever appropriate.

4.1. Bruhat cells and double Bruhat cells for GLn . Our ﬁrst object of inter- est are the double Bruhat cells. Let us introduce them for the group G = GLn(C). We will need some notation. Let B (resp. B ) be the subgroup of upper-triangular − (resp. lower-triangular) matrices in G.LetW=Snbe the symmetric group acting on the set [1,n]= 1,...,n ; we will think of W as a subgroup of G by identify- { } ing a permutation w with the matrix w =(δi,w(j)). The double cosets BwB and B wB are called Bruhat cells (with respect to B and B , respectively). The group− G− has two Bruhat decompositions into a disjoint union− of Bruhat cells (see, e.g., [1, Section 2.4]):

G = BuB = B vB . − − u W v W [∈ [∈ The double Bruhat cells Gu,v are defined by Gu,v = BuB B vB ; ∩ − − thus G is the disjoint union of all Gu,v for (u, v) W W . As an algebraic variety, a double Bruhat cell ∈Gu,v ×is biregularly isomorphic to a Zariski open subset of an affine space of dimension n + `(u)+`(v), where `(u) is the number of inversions of a permutation u; cf. Theorem 1.1. (In other words, Gu,v is isomorphic, as an algebraic variety, to a subset of Cn+`(u)+`(v) obtained by excluding common zeroes of a finite set of polynomials.) Each Bruhat cell (hence each double Bruhat cell) can be described explicitly by a set of conditions specifying vanishing and nonvanishing of certain minors. Let us denote by ∆I,J the minor with the row set I and the column set J;hereIand J are two subsets of the same size in [1,n], and the minor is viewed as a (regular) function on G. (This notation corresponds to that of Definition 1.4, as follows:

the function ∆uωi,vωi in (1.10) becomes the minor ∆u([1,i]),v([1,i]).) The following description of Bruhat cells is probably the most “economical”. Proposition 4.1. Amatrixx Gbelongs to the Bruhat cell BwB if and only if it satisﬁes the following conditions:∈ ∆ =0, for i =1,...,n 1; • w([1,i]),[1,i] 6 − ∆w([1,i 1] j ),[1,i] =0, for all (i, j) such that 1 i

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This proposition can be proved by specializing Propositions 2.9 and 2.6 and Corollary 2.5. Notice that in our present situation the subgroup N (w) (cf. (2.30)) − consists of all unipotent lower-triangular matrices y such that yij = 0 whenever w(i) >w(j). T The transpose map x x transforms a minor ∆I,J into ∆J,I and sends a 7→1 Bruhat cell BwB to B w− B . Thus Proposition 4.1 implies a similar description of the opposite Bruhat− cells B− wB . Combining the two sets of conditions yields an explicit description of the double− − Bruhat cells.

4.2. Factorization problem for GLn . In the situation under consideration, the maximal torus H = B B in G is the subgroup of invertible diagonal matrices. ∩ − n Thus H is naturally identified with C=0 by taking the diagonal entries as coordinates. This allows us to state the factorization6 problem of Section 1.2 in a more symmetric form, as follows. Let Ei,j denote the n n matrix whose (i, j)-entry is equal to 1 while all other entries are 0; let I G denote× the identity matrix. For i =1,...,n 1, let ∈ − 1 00 0 ··· ···  ···0 ··· ···1 ···t ··· ···0  (4.1) xi(t)=I+tEi,i+1 = ··· ···  0 01 0   ··· ···     ··· ··· ··· ··· ··· ···   0 00 1   ··· ···  and   1 00 0 ··· ···  ···0 ··· ···10 ··· ··· ···0  (4.2) x¯(t)=I+tEi+1,i = ··· ··· . i  0 t 1 0   ··· ···     ··· ··· ··· ··· ··· ···   0 00 1   ··· ···  Also, for i =1,...,n and t =0,let  6 1 00 0 ··· ···  ···0 ··· ···t ···0 ··· ···0  (4.3) xi(t)=I+(t 1)Ei,i = ··· ··· . −  0 01 0   ··· ···     ··· ··· ··· ··· ··· ···   0 00 1   ··· ···  The matrices defined in (4.1)–(4.3) are called elementary Jacobi matrices .Itiseasy to see that these matrices generate G as a group. Consider the alphabet of 3n 2 symbols − (4.4) = 1,...,n 1, 1 ,..., n,1,...,n 1 . A { − − } The formulas (4.1)–(4.3) associate a matrix x (t) G to any symbol i and i ∈ ∈A any t C=0 . An analogue of the product map (1.3) is now defined as follows: to ∈ 6 l any sequence i =(i1,...,il) of symbols in , we associate the map xi : C=0 G defined by A 6 →

(4.5) x (t1,...,t )=x (t1) x (t ) . i l i1 ··· il l

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(Thus the diﬀerence with (1.3) is that now the factor a H is split into elementary factors, which are allowed to be spread along the factorization.)∈ For instance, the sequence i = 1 1¯ 21givesrisetothemap

t1 0 10 10 1 t4 t1 t1t4 (4.6) (t1,t2,t3,t4) = . 7→ 01 t2 1 0t3 01 t2 t2t4+t3 The matrix x = xi(t1,...,tl) has a simple combinatorial description in terms of planar networks. This description (cf. [6] and references therein) generalizes the one in [3, Section 2.4], and provides combinatorial formulas for the minors of x as polynomials with nonnegative coeﬃcients in the variables t1,...,tl . The planar network Γ(i) associated to a sequence i of symbols from (see Fig- ure 1) is constructed as a concatenation of “elementary” networks thatA correspond to the parameters t1,...,tl (in this order). Each unbarred, barred, or circled entry ik of i corresponds to a fragment of one of the following three kinds, respectively:

r r r r r r @ trk r r tkr r r @@ tk r r r r r r

ir = ir ir = ¯ir ir = ir k k k (a diagonal edge connects horizontal levels i and i+1; in the examples above, i =2). These fragments are the combinatorial equivalents of the elementary matrices (4.1)– (4.3). Each fragment has a distinguished edge whose weight is tk ; all other edges have weight 1. All edges are presumed oriented left-to-right. We number the sources and sinks of the network Γ(i) bottom-to-top, and deﬁne the weight of a path in Γ(i) to be the product of the weights of all edges in the path. One easily checks that the sum of these weights, over all paths that connect a given source i to a given sink j, is nothing but the matrix element xij of x = xi(t1,...,tl). t 4 11 4 @ t t r t rr@4 5 r 3 3 @ 3 @ t t @ t r rrrrr@1 6 @9 t r 2 @ @ 12 2 t @ t t @ t r rr2 rr@7 t rrr10 @13 r 1 @ 8 @ 1 r rrrrr i = 213 3321 1 214 2 1

Figure 1. Planar network

This observation can be generalized. Let us deﬁne the weight of a family of paths in Γ(i) to be the product of the weights of all paths in the family. Then the minors of x are computed as follows.

Proposition 4.2. A minor ∆I,J(x) equals the sum of weights of all families of vertex-disjoint paths in Γ(i) connecting the sources labeled by I with the sinks labeled by J.

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For example, in Figure 1 we have

x21 = t7t8 + t12t13 + t6t9t12t13 and ∆12,12(x)=t8t12(1 + t6t9). We will be especially interested in a particular class of sequences i which we call factorization schemes (they are analogues of double reduced words of Section 1.2).

Deﬁnition 4.3. Let u and v be two permutations in W = Sn .Afactorization scheme of type (u, v) is a word i of length n + `(u)+`(v) in the alphabet which is an arbitrary shuﬄe of three words of the following kind: A a reduced word for v; • a reduced word for u, with all entries barred; • a permutation of the symbols 1 ,..., n . • These three words will be called, respectively, the E-part,theF-part,andthe H-part of a factorization scheme i. For example, let

u = 4312 = s2s3s1s2s1 S4 , (4.7) ∈ v = 4213 = s1s3s2s1 S4 . ∈ Then (4.8) i = 2 1 3 3 321 1 214 2 1

is a factorization scheme of type (u, v). The following result is an analogue of Theorem 1.2. Theorem 4.4. Let u, v W = S , and let l = n+`(u)+`(v). For any factorization ∈ n scheme i of type (u, v), the product map xi given by (4.5) is a biregular isomorphism l u,v between C=0 and a Zariski open subset of the double Bruhat cell G . 6 The factorization problem for GLn can now be formulated as follows: for a given factorization scheme i, ﬁnd explicit formulas for the components tk in terms of the matrix x = xi(t1,...,tl). By Theorem 4.4, each tk is a rational function in the matrix entries of x. For example, if i = 1 1¯ 2 1, so that the map xi is given by (4.6), then the solution to the factorization problem is given by

det(x) x12 (4.9) t1 = x11 ,t2=x21 ,t3= ,t4= . x11 x11

4.3. The twist maps for GLn . As in the general case, our solution to the factor- u,v ization problem for G = GLn will utilize the “twist maps” ζ : x x0, which are defined for any two permutations u and v. The definition (1.12) can7→ be rewritten as T T T T 1 1 1 T 1 (4.10) x0 = d0 [x u]+ u (x )− v− [v− x ] d0− , − where the following notation is used. The matrix d0 is the diagonal n n matrix with diagonal entries 1, 1, 1, 1,.... For a matrix z G, zT stands for× the transpose − − ∈ of z,andz=[z] [z]0[z]+ denotes the Gaussian decomposition of z (also known as the LDU decomposition).− Finally, the matrix w is obtained from a permutation matrix for w by the following modification: an entry is changed from 1 to 1 whenever it has an odd number of nonzero entries lying below and to the left of− it.

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By Theorem 1.6, the right-hand side of (4.10) is well deﬁned for any x Gu,v,and ∈ 1 1 the twist map ζu,v establishes a biregular isomorphism between Gu,v and Gu− ,v− ; 1 1 the inverse isomorphism is ζu− ,v− . We give below a few examples of explicitly computed twist maps.

Example 4.5. Let G = GL2(C)andu=v=wo. Then (cf. Example 1.8)

0 1 u = v 1 = , − 10−

1 1 1 x11x12− x21− x21− x0= 1 1 . x− x det(x) 12 22 −

Example 4.6. Let G = GL3(C)andu=v=wo.Then

x11 ∆12,13 1   x31 x13 x31 ∆12,23 x31     x0= ∆13,12 x33∆12,12 det(x) x32  .  −   x13 ∆23,12 ∆23,12 ∆12,23 ∆23,12         1 x23 ∆23,23     x13 ∆12,23 det(x)      Example 4.7. Let G = GL4(C)andu=v=wo.Thenx0is equal to

x11 ∆12,14 ∆123,134 1   x14x41 x41∆12,34 x41∆123,234 x41      ∆14,12 x44∆12,12 ∆124,124 x42∆123,134 x41∆123,234 x42   − −   x14∆34,12 ∆34,12∆12,34 ∆34,12∆123,234 ∆34,12   .      ∆134,123 x24∆134,123 x14∆234,123 ∆123,123∆34,34 x33 det(x) ∆34,23   − −  x14∆234,123 ∆12,34∆234,123 ∆123,234∆234,123 ∆234,123        1 x24 ∆23,34 ∆234,234    x14 ∆12,34 ∆123,234 det(x)      Example 4.8. Let n =4,u= 4312, and v = 4213 (cf. (4.7)). Then

0010 0001 0001 010− 0 u= ,v1=  0100 − 1000−  −  1000 0010        

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and

x11 ∆12,13 ∆123,123 1   x41 x13 x41 ∆12,23 x41 ∆123,234 x41      ∆14,12 x43∆12,12 ∆124,123 ∆123,123 x42 x42   −   ∆34,12 x13 ∆34,12 ∆12,23 ∆34,12 ∆123,234 ∆34,12  x0 =   .      1 x23 ∆23,23 ∆23,23     x13 ∆12,23 ∆123,234 ∆123,123         x13 ∆13,23 ∆134,234   0   ∆12,23 ∆123,234 det(x)      Note that, in the course of computing the matrix elements of x0 above, one has to take into account the relations

(4.11) x14 =0,x24 =0, ∆234,123 =0 u,v satisfied by the matrix elements of x G . In particular, our computation of x440 used (4.11) in conjunction with Gröbner∈ bases techniques (see, e.g., [7]). 4.4. Double pseudoline arrangements. As an essential new ingredient in our solution to the factorization problem for GLn, we will represent a factorization scheme i geometrically by the corresponding double pseudoline arrangement (or double wiring diagram). This arrangement is obtained by superimposing two arrangements naturally associated to the E-andF-part of i (cf. [3]). To be self-contained, let us recall the definition of a pseudoline arrangement associated to a reduced word. This is best done by an example. Consider v = 4213 S4, together with the reduced decomposition v = s1s3s2s1 (cf. (4.7)). The corresponding∈ pseudoline arrangement is given in Figure 2; to each entry i of i,we associate a crossing at the ith level, counting from the bottom.

@ @@ @ @@ @ @ @@ @@

1321

Figure 2. Pseudoline arrangement for the reduced word 1321

Let us now consider the factorization scheme i deﬁned by (4.8). The E-part of i is 1321, and we already drew the corresponding arrangement. The F -part of i is 23121. To construct the double pseudoline arrangement for i, we superimpose the arrangements for 1321 and 23121, aligning them closely in the vertical direction, and placing the intersections so that tracing them left-to-right would produce the

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same shuﬄe of the two reduced words that appear in i. This results in the double pseudoline arrangement in Figure 3.

1 4 4 @ @ 3 @ 2 @ @@ 3 3@ @ @ 1 @ @ 4 @ @@ @ 2 2@ @ @ @ 2 @ @ 3 @@ @ @@ @1 1 4 213213 121

Figure 3. Double pseudoline arrangement

The double pseudoline arrangement that corresponds to a factorization scheme i is denoted by Arr(i). The two subarrangements of Arr(i) corresponding to the reduced words for v and u are called the E-andF-part of Arr(i), and their crossing points are referred to as E-andF-crossings, respectively. These crossings are in an obvious bijection with the noncircled entries of i. We next label the pseudolines of Arr(i) using the following important convention. The pseudolines of the F -part of Arr(i) are labeled 1 through n bottom-up at the right end of the arrangement (just as in [3]). At the same time, the pseudolines of the E-part are labeled bottom-up at the left end. See Figure 3. Another numbering that we are going to use is the bottom-to-top numbering of the n 1 horizontal strips containing the crossings of the arrangement. We say that the strip− between the jth and (j + 1)st horizontal lines, counting from the bottom, has level j, and all the E-andF-crossings contained in this strip are of level j. Note that arrangement Arr(i) does not depend on the H-part of the factorization scheme i. In order to include the H-part into the picture, we associate with each entry j a bullet placed on the jth horizontal line. The position of a bullet corresponds to the• position of j in i, so that when the arrangement is traced left- to-right, the crossings and bullets appear in the same order as the entries of i that they represent. The resulting “rigged” arrangement is denoted by Arr (i). Figure 4 shows Arr (i) for the factorization scheme (4.8). • •

@ @ @ @ t @ @ @ @ @ @t @ @ @ @ @ @ @ @ @ @ @ @ t@ @ @ @ @ t 2132133 112421

Figure 4. Arrangement Arr (i) •

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EF @@ @ @ @ @ EF@ FE EF @@ @ @@ @ @ @ @ @ EF@ FE EF@ FF @@ @ @@ @ @ @ @ @ @ @ EE EF@ FE EF@ FF @@ @@ @ @@ @ @@ EF

Figure 5. Types of chambers

To make our terminology uniform, we will refer to the bullets in Arr (i)asH- crossings (despite the fact that they are not crossings geometrically).• Thus the total number of crossings in Arr (i)isl=n+l(u)+l(v), and they are associated • with the variables tk in the factorization (4.5). We will occasionally refer to the crossing in Arr (i) associated with a factorization parameter tk by simply saying • “crossing tk ”. The H-crossing lying on the ith horizontal line will also be denoted by di .

4.5. Solution to the factorization problem. Let us fix permutations u, v Sn and a factorization scheme i of type (u, v); in this section, we present our solution∈ to the corresponding factorization problem. As in [3], the combinatorics needed to formulate the answer involves not only the crossings of the arrangement Arr(i) but also its chambers, which can be defined as horizontal segments between consecutive crossings of the same level. More precisely, each horizontal strip with, say, k crossings breaks down into k + 1 chambers (including the ones at the ends of the strip). Two more chambers are located at the bottom and the top of the arrangement. To illustrate, the arrangement in Figure 3 has 14 chambers; in general, there are l +1 of them. We say that a chamber C is of type EF if the left endpoint of C is an E- crossing, while its right endpoint is an F -crossing. Chambers of types EE, FE and FF are defined in a similar way. Figure 5 shows the types of all 14 chambers of the arrangement in Figure 3. Here and in the sequel, we use the following important convention: on each level, there is a fictitious E-crossing at the left border of the arrangement, and a fictitious F -crossing at the right border. These fictitious crossings determine the types of the chambers adjacent to the boundary of Arr(i). For every chamber C in Arr(i), let I(C) denote the set of labels of the lines of the F -part of the arrangement that pass below C. Analogously, J(C) will consist of the labels of lines of the F -part of Arr(i)thatpassbelowC.ThesetsI(C)and J(C) are called chamber sets for the factorization scheme i. Figure 6 shows the chamber sets I(C)andJ(C) for each chamber of the given arrangement. Note that if C is a chamber of level i,thenbothI(C)andJ(C)haveielements. Our constructions will also involve the “big” chambers formed by the E-part and the F -part of a double pseudoline arrangement, taken separately. We will refer to these “big” chambers as E-chambers and F -chambers, respectively. For example, the arrangement in Figure 3 has 9 E-chambers, which are in obvious bijection with the 9 chambers in Figure 2.

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1234,1234 1 4 4 @ @ 3 234,123 @123,123 123,124 2 @ @@ 3 3 @ @ @ 1 34,12 @ 23,12 23,24 @ 12,24 4 @ @@ @ 2 2 @ @ @ @ 2 3,1 3,2 @ 2,2 2,4 @ 1,4 3 @@ @ @@ @ 1 1 4 , ∅ ∅ Figure 6. Chamber sets

For every chamber C of the arrangement Arr(i), we denote

MC =∆I(C),J(C) ; this minor is considered as a regular function on G (with the convention that ∆ , = 1). For example, if C is the rightmost chamber of level 2 in Figure 6, then ∅ ∅ MC =∆12,24 . To each i =1,... ,n we associate a rational function on Gu,v given by

C MC (4.12) Πi = , MC QC0 0 where C runs over all chambers of levelQ i and type FE, while C0 runs over all chambers of level i and type EF. For example, in Figure 6 we have

∆23,12 Π2 = . ∆34,12 ∆23,24

Also, by convention, Π0 =1. Let be a “big” K-chamber of level i,whereKis one of the symbols E and F . Let L beC the other of these symbols (i.e., L = F if K = E,andL=Eif K = F ). We deﬁne ˜ right M MC (4.13) = C0 0 , MC MC QC00 00 where Q C0 runs over all chambers of level i and type LK to the right of ; • C C00 runs over all chambers of level i and type KL to the right of ; • M˜ = M ,whereCis the (“small”) chamber at the right end of C(inside ), • C C C unless K = E and C is stuck to the right border, in which case M˜ =1. Analogously, ˜ M MC (4.14) left = C0 0 , MC MC QC00 00 where Q C0 runs over all chambers of level i and type KL to the left of ; • C C00 runs over all chambers of level i and type LK to the left of ; • M˜ = M ,whereCis the (“small”) chamber at the left end of C (inside ), • C C C unless K = F and C is stuck to the left border, in which case M˜ =1. We are ﬁnally prepared to state our solution to the factorization problem.

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Theorem 4.9. Let i be a factorization scheme of type (u, v), and suppose a matrix u,v x G admits the factorization x = xi(t1,...,tl) with all tk nonzero complex 1 1 ∈ u,v u− ,v− numbers. Let x0 = ζ (x) G denote the matrix obtained from x by the ∈ “twist” (4.10). Then the factorization parameters tk are determined as follows. If t corresponds to the H -crossing d ,then • k i

Πi(x0) (4.15) tk = , Πi 1(x0) −

where Πi and Πi 1 are given by (4.12). − Let tk correspond to an E-orF-crossing of level i, and let , , , be the • four “big” chambers surrounding this crossing, as shown: A B C D

@A @ BC @

Then D

opp(di+1) opp(di) (x0) (x0) (4.16) tk = MA MD , opp(di+1) opp(di) (x0) (x0) MB MC where we refer to the notation of (4.13)–(4.14) as follows: the superscript opp(di) stands for “left” if the H-crossing di is to the right of tk , and for “right” if di is to the left of tk .

Theorem 4.9 is obtained as a specialization of Theorem 1.9, with the help of the following additional commutation relations:

xi(a)x j (b)=xj(b)xi(a),j/ i, i +1 ; ∈{ }

xi(a)xi(b)=xi(b)xi(a/b);

xi(a)xj(b)=xj(b)xi(ab) ,j=i+1; (4.17)

x¯i(a)xj(b)=xj(b)x¯i(a),j/ i, i +1 ; ∈{ }

x¯i(a)xi(b)=xi(b)x¯i(ab);

x¯i(a)xj(b)=xj(b)x¯i(a/b) ,j=i+1.

Example 4.10. To illustrate Theorem 4.9, let us compute the factorization parameter t9 corresponding to the rightmost F -crossing of level 2 in Figure 4. It is given by

right left (x0) (x0) t9 = MA MD , right left (x0) (x0) MB MC

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where right ( )=∆123 124, M A , right left ∆12,24 ∆23,12 ( )=∆23,24 , ( )= , M B M C ∆23,24 ∆34,12 ∆ left( )= 2,2 . M D ∆3,2 Hence

∆2,2 ∆34,12 ∆123,124 (4.18) t9 = (x0) . ∆3,2 ∆12,24 ∆23,12

Substituting the twisted matrix x0 from Example 4.8 into (4.18) and simplifying, we ﬁnally obtain

∆23,12(x)(x43∆12,12(x) ∆124,123(x)) (4.19) t9 = − . x23 ∆24,12(x)∆123,123(x) This answer can be veriﬁed directly using the combinatorial interpretation of minors ∆I,J in terms of planar networks (see Proposition 4.2). From Figure 1 one obtains: ∆23,12 = t3t7t8t9t12 , x43 = t4 ,∆12,12 = t8t12(1+t6t9), ∆124,123 = t4t8t12 , x23 = t6 ,∆24,12 = t4t7t8t9t12 ,∆123,123 = t3t8t12 , implying (4.19). As in Theorem 1.10, formulas (4.15) and (4.16) imply that the factorization parameters t1,...,tl are related by an invertible monomial transformation to the l minors ∆I(C),J(C)(x0) of the twisted matrix x0 that correspond to the chambers of the arrangement Arr(i), with the bottom chamber excluded. The inverse transformation has the following description which can be deduced from (1.21) (since we left the latter formula without proof, the same is true for our next theorem, although it is not hard to give it a direct proof). Theorem 4.11. Formulas (4.15)–(4.16) are equivalent to the following formulas: 1 − (4.20) ∆I(C),J(C)(x0)= tk , where the product is over all tk which correspondY to the following types of crossings: E-crossings to the right of C such that C lies between the lines intersecting • at tk; F -crossings to the left of C such that C lies between the lines intersecting • at tk; H-crossings to the right of C such that C lies above the E-line passing through • tk; H-crossings to the left of C such that C lies above the F -line passing through • tk. 1 1 For example, in Figure 6, ∆3,1(x0)=(t2t6)− ,∆123,124(x0)=(t1t4t8t12)− ,etc.

4.6. Applications to total positivity. In the case G = GLn(C), the definition of the totally nonnegative variety G 0 given in Section 1.3 is modified as follows: G 0 is the multiplicative semigroup generated≥ by elementary Jacobi matrices (cf. (4.1)–≥ (4.3)) xi(t),x¯i(t), and x i (t) with t>0. It is known [17, 24] that this definition of total nonnegativity is equivalent to the classical one: an invertible matrix x G ∈ belongs to G 0 if and only if all minors ∆I,J(x) (in particular, all matrix entries) are nonnegative.≥

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For any two permutations u and v in Sn , the corresponding totally positive variety is deﬁned by u,v u,v G>0 = G 0 G . ≥ ∩ u,v Each factorization scheme of type (u, v) gives rise to a parametrization of G>0 , according to the following analogue of Theorem 1.3.

Theorem 4.12. For any factorization scheme i =(i1,... ,il) of type (u, v),the l u,v corresponding product map xi : C=0 G given by (4.5) restricts to a bijection l u,v 6 → between R>0 and G>0 . The twist map ζu,v deﬁned by (4.10) respects total positivity: it sends totally 1 1 nonnegative matrices in Gu,v to totally nonnegative matrices in Gu− ,v− (cf. The- orem 1.7). Combining this fact with Theorems 4.12 and 4.9 leads to a family of criteria for total positivity, one for each factorization scheme. For a factorization scheme i =(i1,... ,il)oftype(u, v), let F (i)denotethe collection of l minors ∆ 1 ,whereCruns over all chambers of the ar- uI(C),v− J(C) rangement Arr(i), excluding the bottom chamber. (This notation agrees with that 1 of (1.22).) We note that the pair (uI(C),v− J(C)) will correspond in the same wayasabovetoachamberCif we relabel the pseudolines in Arr(i), numbering the F -pseudolines 1 through n bottom-up at the left end, and the E-pseudolines bottom-up at the right end. See Figure 7.

∆1234,1234 4 2 1 @ 4 ∆ ∆134,234@ ∆ 123,234 @ 134,123 3 @ @@ 1 4 @ @ 3 ∆12,23 ∆ @ ∆ ∆ @ 13,23 13,12 @ 34,12 2 @ @@ @ 3 2 ∆ @ @ @ @ 2 1,3 ∆1,2 @ ∆3,2 ∆3,1 @∆4,1 1 @@ @ @@ @ 4 3 1

Figure 7. Minors ∆ 1 (x) uI(C),v− J(C)

Let F (u, v) denote the union of the collections F (i) for all factorization schemes i of type (u, v). The set F (u, v) can be described directly in the following way. A subset I [1,n] is called a w-chamber set if, together with each element j it also contains⊂ every i such that i 0 for any u− -chamber set I and any v-chamber set J; (3) ∆ 1 (x) > 0 for any chamber C of the arrangement Arr(i) . uI(C),v− J(C) For instance, in our running example where u, v,andiare given by (4.7) and (4.8), a matrix x Gu,v is totally nonnegative if and only if the 13 minors appearing in Figure 7 are all∈ positive if evaluated at x.

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Specializing Theorem 4.13 to the case u = v = wo , we see that the totally wo,wo positive variety G>0 is the classical variety G>0 of the totally positive n n matrices, i.e., those matrices whose all minors are (strictly) positive. Condition× (3) of Theorem 4.13 provides a family of criteria for total positivity, each of which says that a matrix x is totally positive if and only if some collection of n2 minors are positive at x. Diﬀerent factorization schemes i and i0 of the same type (u, v) can have the same collections of chamber sets, thus leading to the same criteria for total positivity. We will say that i and i0 (and the corresponding arrangements Arr(i) and Arr(i0)) are isotopic if they can be obtained from each other by a sequence of the following “trivial 2-moves”:

ij ji , i j 2, ··· ··· ∼ ··· ··· | − |≥ ij ji , i j 2, ··· ··· ∼ ··· ··· | − |≥ ij ji ,i=j, (4.21) ··· ··· ∼ ··· ··· 6 ij j i , ··· ··· ∼ ··· ··· i j j i , ··· ··· ∼ ··· ··· i j j i . ··· ··· ∼ ··· ···

It is not hard to show that i and i0 have the same collection of chamber sets (I(C),J(C)) if and only if they are isotopic. Thus total positivity criteria in The- orem 4.13 are in a bijection with “isotopy types” of arrangements of type (u, v). The set of all isotopy types of arrangements of type (u, v) has a natural structure of a graph deﬁned as follows. We call two isotopy types adjacent if the corresponding collections of chamber sets are obtained from each other by exchanging a single pair (I(C),J(C)) with another one. The graph obtained this way is always connected, and its study is an interesting combinatorial problem. One can check that the adjacency relation in this graph corresponds to the following 3-moves and mixed 2-moves on double reduced words:

iji jij , i j =1, ··· ··· ··· ··· | − | (4.22) iji jij , i j =1, ··· ··· ··· ··· | − | i i ii ··· ··· ··· ··· (cf. Sections 3.3 and 3.4); the connectedness property follows from Proposition 3.7. For G = GL2 and u = v = wo , there are 2 isotopy types. The corresponding collections F (i) are x11,x12,x21, det(x) and x22,x12,x21, det(x) . { } { } In the case of GL3 and u = v = wo , there are 34 isotopy types, giving rise to 34 diﬀerent total positivity criteria. Each of these criteria involves 9 minors. Five of them—the minors

x31 ,x13 , ∆23,12 , ∆12,23 , det(x)

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—are common to all 34 criteria; they correspond to the “unbounded” chambers lying on the periphery of each arrangement. The other four minors that distinguish isotopy types from each other correspond to the bounded chambers. Figure 8 shows a graph with 34 vertices labeled by the quadruples of “bounded” minors that appear in the corresponding total positivity criteria. For an arbitrary n (and u = v = wo ), one obtains various nice (and surprising) total positivity criteria in GLn by making particular choices of (the isotopy type of) a double pseudoline arrangement in Theorem 4.13. Let us discuss two criteria obtained in this way. A minor ∆I,J is called solid if both I and J consist of several consecutive indices. A criterion due to Fekete [10] (see also [12, p. 299]) asserts that (strict) total positivity of a matrix is equivalent to the positivity of all its solid minors. Each of the two criteria described below will strengthen this result. We will consider two factorization schemes of type (wo,wo)havingthesameE- and F -parts (albeit shuffled in a different way). For both of them, the E-part is the lexicographically minimal reduced word for wo, i.e., the reduced word 1, 2, 1, 3, 2, 1,...,n 1,n 2,...,1; − − the F -part is the same but with barred entries. Let i1 denote the shuffle of these parts such that all the unbarred entries precede the barred ones. Let i2 denote the shuffle of the same parts such that every unbarred entry is immediately followed by the corresponding barred entry (so that i2 starts with 1, 1, 2, 2,...). A direct check shows that the corresponding collections of minors F (i1)andF(i2) are given as follows:

F (i1) consists of solid minors ∆ such that 1 I J; • I,J ∈ ∪ F (i2) consists of solid minors ∆ such that min(I)+max(J) n, n +1 . • I,J ∈{ } Each of these two collections consists of n2 minors; and by Theorem 4.13, each of them provides a total positivity criterion that strengthens the one of Fekete’s: a square matrix is totally positive if and only if all the minors in F (i1) (respectively, F (i2)) are positive. It should be mentioned that the first of these criteria was (implicitly) obtained by Cryer [8, Theorems 1.1 and 3.1] using a result of Karlin [16, p. 85]; an explicit statement appears in [13, Theorem 4.1]. The second criterion seems to be new. The equivalence of conditions (2) and (3) in Theorem 4.13 has the following algebraic explanation (cf. Theorem 1.12). Theorem 4.14. For any factorization scheme i of type (u, v), the collection of minors F (i) is a totally positive base (cf. Definition 2.21) for the collection F (u, v). The most significant part of this theorem is that every minor from F (u, v)canbe written as a subtraction-free expression in the minors from F (i).Suchanexpres- sion can be found in a constructive way. To do this, it will be enough to consider two arrangements Arr(i) and Arr(i0) whose isotopy types are adjacent in the graph that we described above; recall that this means that the collection of minors F (i0) is obtained from F (i) by exchanging a single minor ∆ with another minor ∆0.It suffices to show that ∆0 can be written as a subtraction-free expression in the minors from F (i). This can be done with the help of certain 3-term determinantal identities. These identities are stated in the following proposition, which is a specialization of Theorems 1.16 and 1.17. We use the notation Li, Lij, etc., as a shorthand for L i ,L i, j ,etc. ∪{ } ∪{ }

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gABC J tJ J J J J J gBCD J ¨H ¨ H J ¨ tH J ¨¨ HH ¨ H J H¨gBDF gCDE¨H H ¨ J J ttHaBCD ¨ J HH¨H¨¨ J ¨¨H¨HH J J ¨ H¨tH ¨ gDEF H J J ¨ H HaBDF t¨ aCDEJ J H ¨ J ttH ¨ J HH ¨¨ J J H¨aDEF J J J egBF acBFt abCE fgCEJ J J J @ J ttJ J aEF G ttJ @J J @ J @J J t@ J J J @J egAB J ceBFJ acF G @ abEG J bfCE fgAC@J J@ J @ J J @ @ tJ ttJ t @ tJ J t @ J J J J @ abcG J@ J J J @ J J J J ceAB ceF G t bfEG bfAC J J t t bcdG t t J J ¨H ¨¨ HH J J ¨ t H J J ¨¨ HH HcdeG ¨ bdfG J J H ¨ J J t H ¨HbcdA ¨ t HH¨H¨¨ J J ¨H¨H J ¨¨ tdefGHH J ¨ H J J¨cdeA H Notation: H t bdfA¨ J H ¨ t H ¨ t Minors corresponding J HH ¨¨ J H¨ to unbounded chambers a = x11 A =∆23,23 defA J (appear in all criteria): b = x12 B =∆23,13 J t c = x C =∆ 21 13,23 J x13 d = x D =∆ J 22 13,13 x31 e = x E =∆ J 23 13,12 ∆12,23 f = x F =∆ J 32 12,13 J ∆23,12 g = x33 G =∆12,12 J efgA det(x) t

Figure 8. Total positivity criteria for GL3

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Proposition 4.15. For any i, j, k, p [1,n] and I,L [1,n] such that i

For any i, i0,j,j0 1,...,n and I,J 1,...,n such that i

(4.24) ∆Ii,Jj∆Ii0,Jj0 =∆Ii,Jj0 ∆Ii0,Jj +∆I,J∆Iii0,Jjj0 . The identities (4.23)–(4.24) are well known, although their attribution is com- plicated. As early as in 1819 they were proved by P. Desnanot (see [20, pp. 140- 142]). Identities (4.23) are special cases of the (Grassmann-)Pl¨ucker relations (see, e.g., [11, (15.53)]), while identity (4.24) plays a crucial role in C. L. Dodgson’s con- densation method, and because of that is occasionally associated with the name of Lewis Carroll. It would be interesting to see which other classical determinantal identities can

be generalized to the functions ∆uωi,vωi on any semisimple group. We conclude the paper by mentioning one challenging problem of this kind: ﬁnd a generalization of the classical Binet-Cauchy formula for the minors of the product of two matrices:

∆I,J(xy)= ∆I,K(x)∆K,J (y) . XK At present, we only know such a generalization for the minuscule fundamental weights ωi . Acknowledgments. Part of this paper was written when the authors were participating in the special program “Combinatorics” at MSRI in Berkeley in Spring 1997. The second author (A.Z.) gratefully acknowledges the hospitality of his colleagues in Buenos Aires (Alicia Dickenstein and Fernando Cukierman) and Strasbourg (Pe- ter Littelmann and Olivier Mathieu), where he worked on parts of this paper; these visits were supported by the University of Buenos Aires and CNRS, France. A large part of our computations in Section 4 were performed with Maple.

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Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139 E-mail address: [email protected] Department of Mathematics, Northeastern University, Boston, Massachusetts 02115 E-mail address: [email protected]

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